A Brief History of Time - Stephen Hawking... Chapter 6 Figure 6:3 For this mechanism to work, the unseen object has to be very small, like a white dwarf, neutron star, or black hole. From the observed orbit of the visible star, one can determine the lowest possible mass of the unseen object. In the case of Cygnus X-l, this is about six times the mass of the sun, which, according to Chandrasekhar’r result, is too great for the unseen object to be a white dwarf. It is also too large a mass to be a neutron star. It seems, therefore, that it must be a black hole. There are other models to explain Cygnus X-1 that do not include a black hole, but they are all rather far-fetched. A black hole seems to be the only really natural explanation of the observations. Despite this, I had a bet with Kip Thorne of the California Institute of Technology that in fact Cygnus X-1 does not contain a black hole! This was a form f insurance policy for me. I have done a lot of work on black holes, and it would all be wasted if it turned out that black holes do not exist. But in that case, I would have the consolation of winning my bet, which would bring me four years of the magazine Private Eye. In fact, although the situation with Cygnus X-1 has not changed much since we made the bet in 1975, there is now so much other observational evidence in favor of black holes that I have conceded the bet. I paid the specified penalty, which was a one-year subscription to Penthouse, to the outrage of Kip’s liberated wife. We also now have evidence for several other black holes in systems like Cygnus X-1 in our galaxy and in two neighboring galaxies called the Magellanic Clouds. The number of black holes, however, is almost certainly very much higher; in the long history of the universe, many stars must have burned all their nuclear fuel and have had to collapse. The number of black holes may well be greater even than the number of visible stars, which totals about a hundred thousand million in our galaxy alone. The extra gravitational attraction of such a large number of black holes could explain why our galaxy rotates at the rate it does: the mass of the visible stars is insufficient to account for this. We also file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/e.html (8 of 9) [2/20/2001 3:15:08 AM]
A Brief History of Time - Stephen Hawking... Chapter 6 have some evidence that there is a much larger black hole, with a mass of about a hundred thousand times that of the sun, at the center of our galaxy. Stars in the galaxy that come too near this black hole will be torn apart by the difference in the gravitational forces on their near and far sides. Their remains and gas that is thrown off other stars, will fall toward the black hole. As in the case of Cygnus X-l, the gas will spiral inward and will heat up, though not as much as in that case. It will not get hot enough to emit X rays, but it could account for the very compact source of radio waves and infrared rays that is observed at the galactic center. It is thought that similar but even larger black holes, with masses of about a hundred million times the mass of the sun, occur at the centers of quasars. For example, observations with the Hubble telescope of the galaxy known as M87 reveal that it contains a disk of gas 130 light-years across rotating about a central object two thousand million times the mass of the sun. This can only be a black hole. Matter falling into such a supermassive black hole would provide the only source of power great enough to explain the enormous amounts of energy that these objects are emitting. As the matter spirals into the black hole, it would make the black hole rotate in the same direction, causing it to develop a magnetic field rather like that of the earth. Very high-energy particles would be generated near the black hole by the in-falling matter. The magnetic field would be so strong that it could focus these particles into jets ejected outward along the axis of rotation of the black hole, that is, in the directions of its north and south poles. Such jets are indeed observed in a number of galaxies and quasars. One can also consider the possibility that there might be black holes with masses much less than that of the sun. Such black holes could not be formed by gravitational collapse, because their masses are below the Chandrasekhar mass limit: stars of this low mass can support themselves against the force of gravity even when they have exhausted their nuclear fuel. Low-mass black holes could form only if matter was compressed to enormous densities by very large external pressures. Such conditions could occur in a very big hydrogen bomb: the physicist John Wheeler once calculated that if one took all the heavy water in all the oceans of the world, one could build a hydrogen bomb that would compress matter at the center so much that a black hole would be created. (Of course, there would be no one left to observe it!) A more practical possibility is that such low-mass black holes might have been formed in the high temperatures and pressures of the very early universe. Black holes would have been formed only if the early universe had not been perfectly smooth and uniform, because only a small region that was denser than average could be compressed in this way to form a black hole. But we know that there must have been some irregularities, because otherwise the matter in the universe would still be perfectly uniformly distributed at the present epoch, instead of being clumped together in stars and galaxies. Whether the irregularities required to account for stars and galaxies would have led to the formation of a significant number of “primordial” black holes clearly depends on the details of the conditions in the early universe. So if we could determine how many primordial black holes there are now, we would learn a lot about the very early stages of the universe. Primordial black holes with masses more than a thousand million tons (the mass of a large mountain) could be detected only by their gravitational influence on other, visible matter or on the expansion of the universe. However, as we shall learn in the next chapter, black holes are not really black after all: they glow like a hot body, and the smaller they are, the more they glow. So, paradoxically, smaller black holes might actually turn out to be easier to detect than large ones! PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/e.html (9 of 9) [2/20/2001 3:15:08 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 CHAPTER 7 BLACK HOLES AIN’T SO BLACK Before 1970, my research on general relativity had concentrated mainly on the question of whether or not there had been a big bang singularity. However, one evening in November that year, shortly after the birth of my daughter, Lucy, I started to think about black holes as I was getting into bed. My disability makes this rather a slow process, so I had plenty of time. At that date there was no precise definition of which points in space-time lay inside a black hole and which lay outside. I had already discussed with Roger Penrose the idea of defining a black hole as the set of events from which it was not possible to escape to a large distance, which is now the generally accepted definition. It means that the boundary of the black hole, the event horizon, is formed by the light rays that just fail to escape from the black hole, hovering forever just on the edge Figure 7:1. It is a bit like running away from the police and just managing to keep one step ahead but not being able to get clear away! Figure 7:1 Suddenly I realized that the paths of these light rays could never approach one another. If they did they must eventually run into one another. It would be like meeting someone else running away from the police in the opposite direction – you would both be caught! (Or, in this case, fall into a black hole.) But if these light rays were swallowed up by the black hole, then they could not have been on the boundary of the black hole. So the paths of light rays in the file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (1 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 event horizon had always to be moving parallel to, or away from, each other. Another way of seeing this is that the event horizon, the boundary of the black hole, is like the edge of a shadow – the shadow of impending doom. If you look at the shadow cast by a source at a great distance, such as the sun, you will see that the rays of light in the edge are not approaching each other. If the rays of light that form the event horizon, the boundary of the black hole, can never approach each other, the area of the event horizon might stay the same or increase with time, but it could never decrease because that would mean that at least some of the rays of light in the boundary would have to be approaching each other. In fact, the area would increase whenever matter or radiation fell into the black hole Figure 7:2. Figures 7:2 & 7:3 Or if two black holes collided and merged together to form a single black hole, the area of the event horizon of the final black hole would be greater than or equal to the sum of the areas of the event horizons of the original black holes Figure 7:3. This nondecreasing property of the event horizon’s area placed an important restriction on the possible behavior of black holes. I was so excited with my discovery that I did not get much sleep that night. The next day I rang up Roger Penrose. He agreed with me. I think, in fact, that he had been aware of this property of the area. However, he had been using a slightly different definition of a black hole. He had not realized that the boundaries of the black file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (2 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 hole according to the two definitions would be the same, and hence so would their areas, provided the black hole had settled down to a state in which it was not changing with time. The nondecreasing behavior of a black hole’s area was very reminiscent of the behavior of a physical quantity called entropy, which measures the degree of disorder of a system. It is a matter of common experience that disorder will tend to increase if things are left to themselves. (One has only to stop making repairs around the house to see that!) One can create order out of disorder (for example, one can paint the house), but that requires expenditure of effort or energy and so decreases the amount of ordered energy available. A precise statement of this idea is known as the second law of thermodynamics. It states that the entropy of an isolated system always increases, and that when two systems are joined together, the entropy of the combined system is greater than the sum of the entropies of the individual systems. For example, consider a system of gas molecules in a box. The molecules can be thought of as little billiard balls continually colliding with each other and bouncing off the walls of the box. The higher the temperature of the gas, the faster the molecules move, and so the more frequently and harder they collide with the walls of the box and the greater the outward pressure they exert on the walls. Suppose that initially the molecules are all confined to the left-hand side of the box by a partition. If the partition is then removed, the molecules will tend to spread out and occupy both halves of the box. At some later time they could, by chance, all be in the right half or back in the left half, but it is overwhelmingly more probable that there will be roughly equal numbers in the two halves. Such a state is less ordered, or more disordered, than the original state in which all the molecules were in one half. One therefore says that the entropy of the gas has gone up. Similarly, suppose one starts with two boxes, one containing oxygen molecules and the other containing nitrogen molecules. If one joins the boxes together and removes the intervening wall, the oxygen and the nitrogen molecules will start to mix. At a later time the most probable state would be a fairly uniform mixture of oxygen and nitrogen molecules throughout the two boxes. This state would be less ordered, and hence have more entropy, than the initial state of two separate boxes. The second law of thermodynamics has a rather different status than that of other laws of science, such as Newton's law of gravity, for example, because it does not hold always, just in the vast majority of cases. The probability of all the gas molecules in our first box found in one half of the box at a later time is many millions of millions to one, but it can happen. However, if one has a black hole around there seems to be a rather easier way of violating the second law: just throw some matter with a lot of entropy such as a box of gas, down the black hole. The total entropy of matter outside the black hole would go down. One could, of course, still say that the total entropy, including the entropy inside the black hole, has not gone down - but since there is no way to look inside the black hole, we cannot see how much entropy the matter inside it has. It would be nice, then, if there was some feature of the black hole by which observers outside the black hole could tell its entropy, and which would increase whenever matter carrying entropy fell into the black hole. Following the discovery, described above, that the area of the event horizon increased whenever matter fell into a black hole, a research student at Princeton named Jacob Bekenstein suggested that the area of the event horizon was a measure of the entropy of the black hole. As matter carrying entropy fell into a black hole, the area of its event horizon would go up, so that the sum of the entropy of matter outside black holes and the area of the horizons would never go down. This suggestion seemed to prevent the second law of thermodynamics from being violated in most situations. However, there was one fatal flaw. If a black hole has entropy, then it ought to also have a temperature. But a body with a particular temperature must emit radiation at a certain rate. It is a matter of common experience that if one heats up a poker in a fire it glows red hot and emits radiation, but bodies at lower temperatures emit radiation too; one just does not normally notice it because the amount is fairly small. This radiation is required in order to prevent violation of the second law. So black holes ought to emit radiation. But by their very definition, black holes are objects that are not supposed to emit anything. It therefore seemed that the area of the event horizon of a black hole could not be regarded as its entropy. In 1972 I wrote a paper with Brandon Carter and an American colleague, Jim Bardeen, in which we pointed out that although there were many similarities between entropy and the area of the event horizon, there was this apparently fatal difficulty. I must admit that in writing this paper I was motivated partly by irritation with Bekenstein, who, I felt, had misused my discovery of the increase of the area of the event horizon. However, it turned out in the end that he was basically correct, though in a manner he had certainly not expected. In September 1973, while I was visiting Moscow, I discussed black holes with two leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. They convinced me that, according to the quantum mechanical uncertainty principle, rotating black holes should create and emit particles. I believed their arguments on physical grounds, but I did not like the mathematical way in which they calculated the emission. I therefore set about devising a better mathematical treatment, which I described at an informal seminar in Oxford at the end of November 1973. At that time I had not done the calculations to find out how much would actually be emitted. I was expecting to discover just the radiation that Zeldovich and Starobinsky had predicted from rotating black holes. However, when I did the calculation, I file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (3 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 found, to my surprise and annoyance, that even non-rotating black holes should apparently create and emit particles at a steady rate. At first I thought that this emission indicated that one of the approximations I had used was not valid. I was afraid that if Bekenstein found out about it, he would use it as a further argument to support his ideas about the entropy of black holes, which I still did not like. However, the more I thought about it, the more it seemed that the approximations really ought to hold. But what finally convinced me that the emission was real was that the spectrum of the emitted particles was exactly that which would be emitted by a hot body, and that the black hole was emitting particles at exactly the correct rate to prevent violations of the second law. Since then the calculations have been repeated in a number of different forms by other people. They all confirm that a black hole ought to emit particles and radiation as if it were a hot body with a temperature that depends only on the black hole’s mass: the higher the mass, the lower the temperature. How is it possible that a black hole appears to emit particles when we know that nothing can escape from within its event horizon? The answer, quantum theory tells us, is that the particles do not come from within the black hole, but from the “empty” space just outside the black hole’s event horizon! We can understand this in the following way: what we think of as “empty” space cannot be completely empty because that would mean that all the fields, such as the gravitational and electromagnetic fields, would have to be exactly zero. However, the value of a field and its rate of change with time are like the position and velocity of a particle: the uncertainty principle implies that the more accurately one knows one of these quantities, the less accurately one can know the other. So in empty space the field cannot be fixed at exactly zero, because then it would have both a precise value (zero) and a precise rate of change (also zero). There must be a certain minimum amount of uncertainty, or quantum fluctuations, in the value of the field. One can think of these fluctuations as pairs of particles of light or gravity that appear together at some time, move apart, and then come together again and annihilate each other. These particles are virtual particles like the particles that carry the gravitational force of the sun: unlike real particles, they cannot be observed directly with a particle detector. However, their indirect effects, such as small changes in the energy of electron orbits in atoms, can be measured and agree with the theoretical predictions to a remarkable degree of accuracy. The uncertainty principle also predicts that there will be similar virtual pairs of matter particles, such as electrons or quarks. In this case, however, one member of the pair will be a particle and the other an antiparticle (the antiparticles of light and gravity are the same as the particles). Because energy cannot be created out of nothing, one of the partners in a particle/antiparticle pair will have positive energy, and the other partner negative energy. The one with negative energy is condemned to be a short-lived virtual particle because real particles always have positive energy in normal situations. It must therefore seek out its partner and annihilate with it. However, a real particle close to a massive body has less energy than if it were far away, because it would take energy to lift it far away against the gravitational attraction of the body. Normally, the energy of the particle is still positive, but the gravitational field inside a black hole is so strong that even a real particle can have negative energy there. It is therefore possible, if a black hole is present, for the virtual particle with negative energy to fall into the black hole and become a real particle or antiparticle. In this case it no longer has to annihilate with its partner. Its forsaken partner may fall into the black hole as well. Or, having positive energy, it might also escape from the vicinity of the black hole as a real particle or antiparticle Figure 7:4. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (4 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 Figure 7:4 To an observer at a distance, it will appear to have been emitted from the black hole. The smaller the black hole, the shorter the distance the particle with negative energy will have to go before it becomes a real particle, and thus the greater the rate of emission, and the apparent temperature, of the black hole. The positive energy of the outgoing radiation would be balanced by a flow of negative energy particles into the black hole. By Einstein’s equation E = mc2 (where E is energy, m is mass, and c is the speed of light), energy is proportional to mass. A flow of negative energy into the black hole therefore reduces its mass. As the black hole loses mass, the area of its event horizon gets smaller, but this decrease in the entropy of the black hole is more than compensated for by the entropy of the emitted radiation, so the second law is never violated. Moreover, the lower the mass of the black hole, the higher its temperature. So as the black hole loses mass, its temperature and rate of emission increase, so it loses mass more quickly. What happens when the mass of the black hole eventually becomes extremely small is not quite clear, but the most reasonable guess is that it would disappear completely in a tremendous final burst of emission, equivalent to the explosion of millions of H-bombs. A black hole with a mass a few times that of the sun would have a temperature of only one ten millionth of a degree above absolute zero. This is much less than the temperature of the microwave radiation that fills the universe (about file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (5 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 2.7º above absolute zero), so such black holes would emit even less than they absorb. If the universe is destined to go on expanding forever, the temperature of the microwave radiation will eventually decrease to less than that of such a black hole, which will then begin to lose mass. But, even then, its temperature would be so low that it would take about a million million million million million million million million million million million years (1 with sixty-six zeros after it) to evaporate completely. This is much longer than the age of the universe, which is only about ten or twenty thousand million years (1 or 2 with ten zeros after it). On the other hand, as mentioned in Chapter 6, there might be primordial black holes with a very much smaller mass that were made by the collapse of irregularities in the very early stages of the universe. Such black holes would have a much higher temperature and would be emitting radiation at a much greater rate. A primordial black hole with an initial mass of a thousand million tons would have a lifetime roughly equal to the age of the universe. Primordial black holes with initial masses less than this figure would already have completely evaporated, but those with slightly greater masses would still be emitting radiation in the form of X rays and gamma rays. These X rays and gamma rays are like waves of light, but with a much shorter wavelength. Such holes hardly deserve the epithet black: they really are white hot and are emitting energy at a rate of about ten thousand megawatts. One such black hole could run ten large power stations, if only we could harness its power. This would be rather difficult, however: the black hole would have the mass of a mountain compressed into less than a million millionth of an inch, the size of the nucleus of an atom! If you had one of these black holes on the surface of the earth, there would be no way to stop it from falling through the floor to the center of the earth. It would oscillate through the earth and back, until eventually it settled down at the center. So the only place to put such a black hole, in which one might use the energy that it emitted, would be in orbit around the earth – and the only way that one could get it to orbit the earth would be to attract it there by towing a large mass in front of it, rather like a carrot in front of a donkey. This does not sound like a very practical proposition, at least not in the immediate future. But even if we cannot harness the emission from these primordial black holes, what are our chances of observing them? We could look for the gamma rays that the primordial black holes emit during most of their lifetime. Although the radiation from most would be very weak because they are far away, the total from all of them might be detectable. We do observe such a background of gamma rays: Figure 7:5 shows how the observed intensity differs at different frequencies (the number of waves per second). However, this background could have been, and probably was, generated by processes other than primordial black holes. The dotted line in Figure 7:5 shows how the intensity should vary with frequency for gamma rays given off by primordial black holes, if there were on average 300 per cubic light-year. One can therefore say that the observations of the gamma ray background do not provide any positive evidence for primordial black holes, but they do tell us that on average there cannot be more than 300 in every cubic light-year in the universe. This limit means that primordial black holes could make up at most one millionth of the matter in the universe. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (6 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 Figure 7:5 With primordial black holes being so scarce, it might seem unlikely that there would be one near enough for us to observe as an individual source of gamma rays. But since gravity would draw primordial black holes toward any matter, they should be much more common in and around galaxies. So although the gamma ray background tells us that there can be no more than 300 primordial black holes per cubic light-year on average, it tells us nothing about how common they might be in our own galaxy. If they were, say, a million times more common than this, then the nearest black hole to us would probably be at a distance of about a thousand million kilometers, or about as far away as Pluto, the farthest known planet. At this distance it would still be very difficult to detect the steady emission of a black hole, even if it was ten thousand megawatts. In order to observe a primordial black hole one would have to detect several gamma ray quanta coming from the same direction within a reasonable space of time, such as a week. Otherwise, they might simply be part of the background. But Planck’s quantum principle tells us that each gamma ray quantum has a very high energy, because gamma rays have a very high frequency, so it would not take many quanta to radiate even ten thousand megawatts. And to observe these few coming from the distance of Pluto would require a larger gamma ray detector than any that have been constructed so far. Moreover, the detector would have to be in space, because gamma rays cannot penetrate the atmosphere. Of course, if a black hole as close as Pluto were to reach the end of its life and blow up, it would be easy to detect the final burst of emission. But if the black hole has been emitting for the last ten or twenty thousand million years, the chance of it reaching the end of its life within the next few years, rather than several million years in the past or future, is really rather small! So in order to have a reasonable chance of seeing an explosion before your research grant ran out, you would have to find a way to detect any explosions within a distance of about one light-year. In fact bursts of file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (7 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 7 gamma rays from space have been detected by satellites originally constructed to look for violations of the Test Ban Treaty. These seem to occur about sixteen times a month and to be roughly uniformly distributed in direction across the sky. This indicates that they come from outside the Solar System since otherwise we would expect them to be concentrated toward the plane of the orbits of the planets. The uniform distribution also indicates that the sources are either fairly near to us in our galaxy or right outside it at cosmological distances because otherwise, again, they would be concentrated toward the plane of the galaxy. In the latter case, the energy required to account for the bursts would be far too high to have been produced by tiny black holes, but if the sources were close in galactic terms, it might be possible that they were exploding black holes. I would very much like this to be the case but I have to recognize that there are other possible explanations for the gamma ray bursts, such as colliding neutron stars. New observations in the next few years, particularly by gravitational wave detectors like LIGO, should enable us to discover the origin of the gamma ray bursts. Even if the search for primordial black holes proves negative, as it seems it may, it will still give us important information about the very early stages of the universe. If the early universe had been chaotic or irregular, or if the pressure of matter had been low, one would have expected it to produce many more primordial black holes than the limit already set by our observations of the gamma ray background. Only if the early universe was very smooth and uniform, with a high pressure, can one explain the absence of observable numbers of primordial black holes. The idea of radiation from black holes was the first example of a prediction that depended in an essential way on both the great theories of this century, general relativity and quantum mechanics. It aroused a lot of opposition initially because it upset the existing viewpoint: “How can a black hole emit anything?” When I first announced the results of my calculations at a conference at the Rutherford-Appleton Laboratory near Oxford, I was greeted with general incredulity. At the end of my talk the chairman of the session, John G. Taylor from Kings College, London, claimed it was all nonsense. He even wrote a paper to that effect. However, in the end most people, including John Taylor, have come to the conclusion that black holes must radiate like hot bodies if our other ideas about general relativity and quantum mechanics are correct. Thus, even though we have not yet managed to find a primordial black hole, there is fairly general agreement that if we did, it would have to be emitting a lot of gamma rays and X rays. The existence of radiation from black holes seems to imply that gravitational collapse is not as final and irreversible as we once thought. If an astronaut falls into a black hole, its mass will increase, but eventually the energy equivalent of that extra mass will be returned to the universe in the form of radiation. Thus, in a sense, the astronaut will be “recycled.” It would be a poor sort of immortality, however, because any personal concept of time for the astronaut would almost certainly come to an end as he was torn apart inside the black hole! Even the types of particles that were eventually emitted by the black hole would in general be different from those that made up the astronaut: the only feature of the astronaut that would survive would be his mass or energy. The approximations I used to derive the emission from black holes should work well when the black hole has a mass greater than a fraction of a gram. However, they will break down at the end of the black hole’s life when its mass gets very small. The most likely outcome seems to be that the black hole will just disappear, at least from our region of the universe, taking with it the astronaut and any singularity there might be inside it, if indeed there is one. This was the first indication that quantum mechanics might remove the singularities that were predicted by general relativity. However, the methods that I and other people were using in 1974 were not able to answer questions such as whether singularities would occur in quantum gravity. From 1975 onward I therefore started to develop a more powerful approach to quantum gravity based on Richard Feynrnan’s idea of a sum over histories. The answers that this approach suggests for the origin and fate of the universe and its contents, such as astronauts, will be de-scribed in the next two chapters. We shall see that although the uncertainty principle places limitations on the accuracy of all our predictions, it may at the same time remove the fundamental unpredictability that occurs at a space-time singularity. PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/f.html (8 of 8) [2/20/2001 3:15:18 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 CHAPTER 8 THE ORIGIN AND FATE OF THE UNIVERSE Einstein’s general theory of relativity, on its own, predicted that space-time began at the big bang singularity and would come to an end either at the big crunch singularity (if the whole universe recollapsed), or at a singularity inside a black hole (if a local region, such as a star, were to collapse). Any matter that fell into the hole would be destroyed at the singularity, and only the gravitational effect of its mass would continue to be felt outside. On the other hand, when quantum effects were taken into account, it seemed that the mass or energy of the matter would eventually be returned to the rest of the universe, and that the black hole, along with any singularity inside it, would evaporate away and finally disappear. Could quantum mechanics have an equally dramatic effect on the big bang and big crunch singularities? What really happens during the very early or late stages of the universe, when gravitational fields are so strong that quantum effects cannot be ignored? Does the universe in fact have a beginning or an end? And if so, what are they like? Throughout the 1970s I had been mainly studying black holes, but in 1981 my interest in questions about the origin and fate of the universe was reawakened when I attended a conference on cosmology organized by the Jesuits in the Vatican. The Catholic Church had made a bad mistake with Galileo when it tried to lay down the law on a question of science, declaring that the sun went round the earth. Now, centuries later, it had decided to invite a number of experts to advise it on cosmology. At the end of the conference the participants were granted an audience with the Pope. He told us that it was all right to study the evolution of the universe after the big bang, but we should not inquire into the big bang itself because that was the moment of Creation and therefore the work of God. I was glad then that he did not know the subject of the talk I had just given at the conference – the possibility that space-time was finite but had no boundary, which means that it had no beginning, no moment of Creation. I had no desire to share the fate of Galileo, with whom I feel a strong sense of identity, partly because of the coincidence of having been born exactly 300 years after his death! In order to explain the ideas that I and other people have had about how quantum mechanics may affect the origin and fate of the universe, it is necessary first to understand the generally accepted history of the universe, according to what is known as the “hot big bang model.” This assumes that the universe is described by a Friedmann model, right back to the big bang. In such models one finds that as the universe expands, any matter or radiation in it gets cooler. (When the universe doubles in size, its temperature falls by half.) Since temperature is simply a measure of the average energy – or speed – of the particles, this cooling of the universe would have a major effect on the matter in it. At very high temperatures, particles would be moving around so fast that they could escape any attraction toward each other due to nuclear or electromagnetic forces, but as they cooled off one would expect particles that attract each other to start to clump together. Moreover, even the types of particles that exist in the universe would depend on the temperature. At high enough temperatures, particles have so much energy that whenever they collide many different particle/antiparticle pairs would be produced – and although some of these particles would annihilate on hitting antiparticles, they would be produced more rap-idly than they could annihilate. At lower temperatures, however, when colliding particles have less energy, particle/antiparticle pairs would be produced less quickly – and annihilation would become faster than production. At the big bang itself the universe is thought to have had zero size, and so to have been infinitely hot. But as the universe expanded, the temperature of the radiation decreased. One second after the big bang, it would have fallen to about ten thousand million degrees. This is about a thousand times the temperature at the center of the sun, but temperatures as high as this are reached in H-bomb explosions. At this time the universe would have contained mostly photons, electrons, and neutrinos (extremely light particles that are affected only by the weak force and gravity) and their antiparticles, together with some protons and neutrons. As the universe continued to expand and the temperature to drop, the rate at which electron/antielectron pairs were being produced in collisions would have fallen below the rate at which they were being destroyed by annihilation. So most of the electrons and antielectrons would have annihilated with each other to produce more photons, leaving only a few electrons left over. The neutrinos and antineutrinos, however, would not have annihilated with each other, because these particles interact with themselves and with other particles only very weakly. So they should still be around today. If we could observe them, it would provide a good test of this picture of a very hot early stage of the universe. Unfortunately, their energies nowadays would be too low for us to observe them directly. However, if neutrinos are not massless, but have a small mass of their own, as suggested by some recent experiments, we might be able to detect them indirectly: they could be a form of “dark matter,” like that mentioned earlier, with sufficient gravitational attraction to stop the expansion of the universe and cause it to collapse again. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (1 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 About one hundred seconds after the big bang, the temperature would have fallen to one thousand million degrees, the temperature inside the hottest stars. At this temperature protons and neutrons would no longer have sufficient energy to escape the attraction of the strong nuclear force, and would have started to combine together to produce the nuclei of atoms of deuterium (heavy hydrogen), which contain one proton and one neutron. The deuterium nuclei would then have combined with more protons and neutrons to make helium nuclei, which contain two protons and two neutrons, and also small amounts of a couple of heavier elements, lithium and beryllium. One can calculate that in the hot big bang model about a quarter of the protons and neutrons would have been converted into helium nuclei, along with a small amount of heavy hydrogen and other elements. The remaining neutrons would have decayed into protons, which are the nuclei of ordinary hydrogen atoms. This picture of a hot early stage of the universe was first put forward by the scientist George Gamow in a famous paper written in 1948 with a student of his, Ralph Alpher. Gamow had quite a sense of humor – he persuaded the nuclear scientist Hans Bethe to add his name to the paper to make the list of authors “Alpher, Bethe, Gamow,” like the first three letters of the Greek alphabet, alpha, beta, gamma: particularly appropriate for a paper on the beginning of the universe! In this paper they made the remarkable prediction that radiation (in the form of photons) from the very hot early stages of the universe should still be around today, but with its temperature reduced to only a few degrees above absolute zero (–273ºC). It was this radiation that Penzias and Wilson found in 1965. At the time that Alpher, Bethe, and Gamow wrote their paper, not much was known about the nuclear reactions of protons and neutrons. Predictions made for the proportions of various elements in the early universe were therefore rather inaccurate, but these calculations have been repeated in the light of better knowledge and now agree very well with what we observe. It is, moreover, very difficult to explain in any other way why there should be so much helium in the universe. We are therefore fairly confident that we have the right picture, at least back to about one second after the big bang. Within only a few hours of the big bang, the production of helium and other elements would have stopped. And after that, for the next million years or so, the universe would have just continued expanding, without anything much happening. Eventually, once the temperature had dropped to a few thousand degrees, and electrons and nuclei no longer had enough energy to overcome the electromagnetic attraction between them, they would have started combining to form atoms. The universe as a whole would have continued expanding and cooling, but in regions that were slightly denser than average, the expansion would have been slowed down by the extra gravitational attraction. This would eventually stop expansion in some regions and cause them to start to recollapse. As they were collapsing, the gravitational pull of matter outside these regions might start them rotating slightly. As the collapsing region got smaller, it would spin faster – just as skaters spinning on ice spin faster as they draw in their arms. Eventually, when the region got small enough, it would be spinning fast enough to balance the attraction of gravity, and in this way disklike rotating galaxies were born. Other regions, which did not happen to pick up a rotation, would become oval-shaped objects called elliptical galaxies. In these, the region would stop collapsing because individual parts of the galaxy would be orbiting stably round its center, but the galaxy would have no overall rotation. As time went on, the hydrogen and helium gas in the galaxies would break up into smaller clouds that would collapse under their own gravity. As these contracted, and the atoms within them collided with one another, the temperature of the gas would increase, until eventually it became hot enough to start nuclear fusion reactions. These would convert the hydrogen into more helium, and the heat given off would raise the pressure, and so stop the clouds from contracting any further. They would remain stable in this state for a long time as stars like our sun, burning hydrogen into helium and radiating the resulting energy as heat and light. More massive stars would need to be hotter to balance their stronger gravitational attraction, making the nuclear fusion reactions proceed so much more rapidly that they would use up their hydrogen in as little as a hundred million years. They would then contract slightly, and as they heated up further, would start to convert helium into heavier elements like carbon or oxygen. This, however, would not release much more energy, so a crisis would occur, as was described in the chapter on black holes. What happens next is not completely clear, but it seems likely that the central regions of the star would collapse to a very dense state, such as a neutron star or black hole. The outer regions of the star may sometimes get blown off in a tremendous explosion called a supernova, which would outshine all the other stars in its galaxy. Some of the heavier elements produced near the end of the star’s life would be flung back into the gas in the galaxy, and would provide some of the raw material for the next generation of stars. Our own sun contains about 2 percent of these heavier elements, because it is a second- or third-generation star, formed some five thousand million years ago out of a cloud of rotating gas containing the debris of earlier supernovas. Most of the gas in that cloud went to form the sun or got blown away, but a small amount of the heavier elements collected together to form the bodies that now orbit the sun as planets like the earth. The earth was initially very hot and without an atmosphere. In the course of time it cooled and acquired an file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (2 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 atmosphere from the emission of gases from the rocks. This early atmosphere was not one in which we could have survived. It contained no oxygen, but a lot of other gases that are poisonous to us, such as hydrogen sulfide (the gas that gives rotten eggs their smell). There are, however, other primitive forms of life that can flourish under such conditions. It is thought that they developed in the oceans, possibly as a result of chance combinations of atoms into large structures, called macromolecules, which were capable of assembling other atoms in the ocean into similar structures. They would thus have reproduced themselves and multiplied. In some cases there would be errors in the reproduction. Mostly these errors would have been such that the new macromolecule could not reproduce itself and eventually would have been destroyed. However, a few of the errors would have produced new macromolecules that were even better at reproducing themselves. They would have therefore had an advantage and would have tended to replace the original macromolecules. In this way a process of evolution was started that led to the development of more and more complicated, self-reproducing organisms. The first primitive forms of life consumed various materials, including hydrogen sulfide, and released oxygen. This gradually changed the atmosphere to the composition that it has today, and allowed the development of higher forms of life such as fish, reptiles, mammals, and ultimately the human race. This picture of a universe that started off very hot and cooled as it expanded is in agreement with all the observational evidence that we have today. Nevertheless, it leaves a number of important questions unanswered: 1. Why was the early universe so hot? 2. Why is the universe so uniform on a large scale? Why does it look the same at all points of space and in all directions? In particular, why is the temperature of the microwave back-ground radiation so nearly the same when we look in different directions? It is a bit like asking a number of students an exam question. If they all give exactly the same answer, you can be pretty sure they have communicated with each other. Yet, in the model described above, there would not have been time since the big bang for light to get from one distant region to another, even though the regions were close together in the early universe. According to the theory of relativity, if light cannot get from one region to another, no other information can. So there would be no way in which different regions in the early universe could have come to have the same temperature as each other, unless for some unexplained reason they happened to start out with the same temperature. 3. Why did the universe start out with so nearly the critical rate of expansion that separates models that recollapse from those that go on expanding forever, that even now, ten thousand million years later, it is still expanding at nearly the critical rate? If the rate of expansion one second after the big bang had been smaller by even one part in a hundred thousand million million, the universe would have recollapsed before it ever reached its present size. 4. Despite the fact that the universe is so uniform and homogeneous on a large scale, it contains local irregularities, such as stars and galaxies. These are thought to have developed from small differences in the density of the early universe from one region to another. What was the origin of these density fluctuations? The general theory of relativity, on its own, cannot explain these features or answer these questions because of its prediction that the universe started off with infinite density at the big bang singularity. At the singularity, general relativity and all other physical laws would break down: one couldn’t predict what would come out of the singularity. As explained before, this means that one might as well cut the big bang, and any events before it, out of the theory, because they can have no effect on what we observe. Space-time would have a boundary – a beginning at the big bang. Science seems to have uncovered a set of laws that, within the limits set by the uncertainty principle, tell us how the universe will develop with time, if we know its state at any one time. These laws may have originally been decreed by God, but it appears that he has since left the universe to evolve according to them and does not now intervene in it. But how did he choose the initial state or configuration of the universe? What were the “boundary conditions” at the beginning of time? One possible answer is to say that God chose the initial configuration of the universe for reasons that we cannot hope to understand. This would certainly have been within the power of an omnipotent being, but if he had started it off in such an incomprehensible way, why did he choose to let it evolve according to laws that we could understand? The whole history of science has been the gradual realization that events do not happen in an arbitrary manner, but that they reflect a certain underlying order, which may or may not be divinely inspired. It would be only natural to suppose that this order should apply not only to the laws, but also to the conditions at the boundary of space-time that specify the initial state of the universe. There may be a large number of models of the universe with different initial conditions that all obey the laws. There ought to be some principle that picks out one initial state, and hence file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (3 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 one model, to represent our universe. One such possibility is what are called chaotic boundary conditions. These implicitly assume either that the universe is spatially infinite or that there are infinitely many universes. Under chaotic boundary conditions, the probability of finding any particular region of space in any given configuration just after the big bang is the same, in some sense, as the probability of finding it in any other configuration: the initial state of the universe is chosen purely randomly. This would mean that the early universe would have probably been very chaotic and irregular because there are many more chaotic and disordered configurations for the universe than there are smooth and ordered ones. (If each configuration is equally probable, it is likely that the universe started out in a chaotic and disordered state, simply because there are so many more of them.) It is difficult to see how such chaotic initial conditions could have given rise to a universe that is so smooth and regular on a large scale as ours is today. One would also have expected the density fluctuations in such a model to have led to the formation of many more primordial black holes than the upper limit that has been set by observations of the gamma ray background. If the universe is indeed spatially infinite, or if there are infinitely many universes, there would probably be some large regions somewhere that started out in a smooth and uniform manner. It is a bit like the well-known horde of monkeys hammering away on typewriters – most of what they write will be garbage, but very occasionally by pure chance they will type out one of Shakespeare’s sonnets. Similarly, in the case of the universe, could it be that we are living in a region that just happens by chance to be smooth and uniform? At first sight this might seem very improbable, because such smooth regions would be heavily outnumbered by chaotic and irregular regions. However, suppose that only in the smooth regions were galaxies and stars formed and were conditions right for the development of complicated self-replicating organisms like ourselves who were capable of asking the question: why is the universe so smooth.? This is an example of the application of what is known as the anthropic principle, which can be paraphrased as “We see the universe the way it is because we exist.” There are two versions of the anthropic principle, the weak and the strong. The weak anthropic principle states that in a universe that is large or infinite in space and/or time, the conditions necessary for the development of intelligent life will be met only in certain regions that are limited in space and time. The intelligent beings in these regions should therefore not be surprised if they observe that their locality in the universe satisfies the conditions that are necessary for their existence. It is a bit like a rich person living in a wealthy neighborhood not seeing any poverty. One example of the use of the weak anthropic principle is to “explain” why the big bang occurred about ten thousand million years ago – it takes about that long for intelligent beings to evolve. As explained above, an early generation of stars first had to form. These stars converted some of the original hydrogen and helium into elements like carbon and oxygen, out of which we are made. The stars then exploded as supernovas, and their debris went to form other stars and planets, among them those of our Solar System, which is about five thousand million years old. The first one or two thousand million years of the earth’s existence were too hot for the development of anything complicated. The remaining three thousand million years or so have been taken up by the slow process of biological evolution, which has led from the simplest organisms to beings who are capable of measuring time back to the big bang. Few people would quarrel with the validity or utility of the weak anthropic principle. Some, however, go much further and propose a strong version of the principle. According to this theory, there are either many different universes or many different regions of a single universe, each with its own initial configuration and, perhaps, with its own set of laws of science. In most of these universes the conditions would not be right for the development of complicated organisms; only in the few universes that are like ours would intelligent beings develop and ask the question, “Why is the universe the way we see it?” The answer is then simple: if it had been different, we would not be here! The laws of science, as we know them at present, contain many fundamental numbers, like the size of the electric charge of the electron and the ratio of the masses of the proton and the electron. We cannot, at the moment at least, predict the values of these numbers from theory – we have to find them by observation. It may be that one day we shall discover a complete unified theory that predicts them all, but it is also possible that some or all of them vary from universe to universe or within a single universe. The remarkable fact is that the values of these numbers seem to have been very finely adjusted to make possible the development of life. For example, if the electric charge of the electron had been only slightly different, stars either would have been unable to burn hydrogen and helium, or else they would not have exploded. Of course, there might be other forms of intelligent life, not dreamed of even by writers of science fiction, that did not require the light of a star like the sun or the heavier chemical elements that are made in stars and are flung back into space when the stars explode. Nevertheless, it seems clear that there are relatively few ranges of values for the numbers that would allow the development of any form of intelligent life. Most sets of values would give rise to universes that, although they might be very beautiful, would contain no one able to wonder at that beauty. One can take this either as evidence of a divine purpose in Creation and the choice of the file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (4 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 laws of science or as support for the strong anthropic principle. There are a number of objections that one can raise to the strong anthropic principle as an explanation of the observed state of the universe. First, in what sense can all these different universes be said to exist? If they are really separate from each other, what happens in another universe can have no observable consequences in our own universe. We should therefore use the principle of economy and cut them out of the theory. If, on the other hand, they are just different regions of a single universe, the laws of science would have to be the same in each region, because otherwise one could not move continuously from one region to another. In this case the only difference between the regions would be their initial configurations and so the strong anthropic principle would reduce to the weak one. A second objection to the strong anthropic principle is that it runs against the tide of the whole history of science. We have developed from the geocentric cosmologies of Ptolemy and his forebears, through the heliocentric cosmology of Copernicus and Galileo, to the modern picture in which the earth is a medium-sized planet orbiting around an average star in the outer suburbs of an ordinary spiral galaxy, which is itself only one of about a million million galaxies in the observable universe. Yet the strong anthropic principle would claim that this whole vast construction exists simply for our sake. This is very hard to believe. Our Solar System is certainly a prerequisite for our existence, hand one might extend this to the whole of our galaxy to allow for an earlier generation of stars that created the heavier elements. But there does not seem to be any need for all those other galaxies, nor for the universe to be so uniform and similar in every direction on the large scale. One would feel happier about the anthropic principle, at least in its weak version, if one could show that quite a number of different initial configurations for the universe would have evolved to produce a universe like the one we observe. If this is the case, a universe that developed from some sort of random initial conditions should contain a number of regions that are smooth and uniform and are suitable for the evolution of intelligent life. On the other hand, if the initial state of the universe had to be chosen extremely carefully to lead to something like what we see around us, the universe would be unlikely to contain any region in which life would appear. In the hot big bang model described above, there was not enough time in the early universe for heat to have flowed from one region to another. This means that the initial state of the universe would have to have had exactly the same temperature everywhere in order to account for the fact that the microwave back-ground has the same temperature in every direction we look. The initial rate of expansion also would have had to be chosen very precisely for the rate of expansion still to be so close to the critical rate needed to avoid recollapse. This means that the initial state of the universe must have been very carefully chosen indeed if the hot big bang model was correct right back to the beginning of time. It would be very difficult to explain why the universe should have begun in just this way, except as the act of a God who intended to create beings like us. In an attempt to find a model of the universe in which many different initial configurations could have evolved to something like the present universe, a scientist at the Massachusetts Institute of Technology, Alan Guth, suggested that the early universe might have gone through a period of very rapid expansion. This expansion is said to be “inflationary,” meaning that the universe at one time expanded at an increasing rate rather than the decreasing rate that it does today. According to Guth, the radius of the universe increased by a million million million million million (1 with thirty zeros after it) times in only a tiny fraction of a second. Guth suggested that the universe started out from the big bang in a very hot, but rather chaotic, state. These high temperatures would have meant that the particles in the universe would be moving very fast and would have high energies. As we discussed earlier, one would expect that at such high temperatures the strong and weak nuclear forces and the electromagnetic force would all be unified into a single force. As the universe expanded, it would cool, and particle energies would go down. Eventually there would be what is called a phase transition and the symmetry between the forces would be broken: the strong force would become different from the weak and electromagnetic forces. One common example of a phase transition is the freezing of water when you cool it down. Liquid water is symmetrical, the same at every point and in every direction. However, when ice crystals form, they will have definite positions and will be lined up in some direction. This breaks water’s symmetry. In the case of water, if one is careful, one can “supercool” it: that is, one can reduce the temperature below the freezing point (OºC) without ice forming. Guth suggested that the universe might behave in a similar way: the temperature might drop below the critical value without the symmetry between the forces being broken. If this happened, the universe would be in an unstable state, with more energy than if the symmetry had been broken. This special extra energy can be shown to have an antigravitational effect: it would have acted just like the cosmological constant that Einstein introduced into general relativity when he was trying to construct a static model of the universe. Since the universe would already be expanding just as in the hot big bang model, the repulsive effect of file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (5 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 this cosmological constant would therefore have made the universe expand at an ever-increasing rate. Even in regions where there were more matter particles than average, the gravitational attraction of the matter would have been outweighed by the repulsion of the effective cosmological constant. Thus these regions would also expand in an accelerating inflationary manner. As they expanded and the matter particles got farther apart, one would be left with an expanding universe that contained hardly any particles and was still in the supercooled state. Any irregularities in the universe would simply have been smoothed out by the expansion, as the wrinkles in a balloon are smoothed away when you blow it up. Thus the present smooth and uniform state of the universe could have evolved from many different non-uniform initial states. In such a universe, in which the expansion was accelerated by a cosmological constant rather than slowed down by the gravitational attraction of matter, there would be enough time for light to travel from one region to another in the early universe. This could provide a solution to the problem, raised earlier, of why different regions in the early universe have the same properties. Moreover, the rate of expansion of the universe would automatically become very close to the critical rate determined by the energy density of the universe. This could then explain why the rate of expansion is still so close to the critical rate, without having to assume that the initial rate of expansion of the universe was very carefully chosen. The idea of inflation could also explain why there is so much matter in the universe. There are something like ten million million million million million million million million million million million million million million (1 with eighty zeros after it) particles in the region of the universe that we can observe. Where did they all come from? The answer is that, in quantum theory, particles can be created out of energy in the form of particle/antiparticle pairs. But that just raises the question of where the energy came from. The answer is that the total energy of the universe is exactly zero. The matter in the universe is made out of positive energy. However, the matter is all attracting itself by gravity. Two pieces of matter that are close to each other have less energy than the same two pieces a long way apart, because you have to expend energy to separate them against the gravitational force that is pulling them together. Thus, in a sense, the gravitational field has negative energy. In the case of a universe that is approximately uniform in space, one can show that this negative gravitational energy exactly cancels the positive energy represented by the matter. So the total energy of the universe is zero. Now twice zero is also zero. Thus the universe can double the amount of positive matter energy and also double the negative gravitational energy without violation of the conservation of energy. This does not happen in the normal expansion of the universe in which the matter energy density goes down as the universe gets bigger. It does happen, however, in the inflationary expansion because the energy density of the supercooled state remains constant while the universe expands: when the universe doubles in size, the positive matter energy and the negative gravitational energy both double, so the total energy remains zero. During the inflationary phase, the universe increases its size by a very large amount. Thus the total amount of energy available to make particles becomes very large. As Guth has remarked, “It is said that there’s no such thing as a free lunch. But the universe is the ultimate free lunch.” The universe is not expanding in an inflationary way today. Thus there has to be some mechanism that would eliminate the very large effective cosmological constant and so change the rate of expansion from an accelerated one to one that is slowed down by gravity, as we have today. In the inflationary expansion one might expect that eventually the symmetry between the forces would be broken, just as super-cooled water always freezes in the end. The extra energy of the unbroken symmetry state would then be released and would reheat the universe to a temperature just below the critical temperature for symmetry between the forces. The universe would then go on to expand and cool just like the hot big bang model, but there would now be an explanation of why the universe was expanding at exactly the critical rate and why different regions had the same temperature. In Guth’s original proposal the phase transition was supposed to occur suddenly, rather like the appearance of ice crystals in very cold water. The idea was that “bubbles” of the new phase of broken symmetry would have formed in the old phase, like bubbles of steam surrounded by boiling water. The bubbles were supposed to expand and meet up with each other until the whole universe was in the new phase. The trouble was, as I and several other people pointed out, that the universe was expanding so fast that even if the bubbles grew at the speed of light, they would be moving away from each other and so could not join up. The universe would be left in a very non-uniform state, with some regions still having symmetry between the different forces. Such a model of the universe would not correspond to what we see. In October 1981, I went to Moscow for a conference on quantum gravity. After the conference I gave a seminar on the inflationary model and its problems at the Sternberg Astronomical Institute. Before this, I had got someone else to give my lectures for me, because most people could not understand my voice. But there was not time to prepare this seminar, so I gave it myself, with one of my graduate students repeating my words. It worked well, and gave me file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (6 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 much more contact with my audience. In the audience was a young Russian, Andrei Linde, from the Lebedev Institute in Moscow. He said that the difficulty with the bubbles not joining up could be avoided if the bubbles were so big that our region of the universe is all contained inside a single bubble. In order for this to work, the change from symmetry to broken symmetry must have taken place very slowly inside the bubble, but this is quite possible according to grand unified theories. Linde’s idea of a slow breaking of symmetry was very good, but I later realized that his bubbles would have to have been bigger than the size of the universe at the time! I showed that instead the symmetry would have broken everywhere at the same time, rather than just inside bubbles. This would lead to a uniform universe, as we observe. I was very excited by this idea and discussed it with one of my students, Ian Moss. As a friend of Linde’s, I was rather embarrassed, however, when I was later sent his paper by a scientific journal and asked whether it was suitable for publication. I replied that there was this flaw about the bubbles being bigger than the universe, but that the basic idea of a slow breaking of symmetry was very good. I recommended that the paper ¿ published as it was because it would take Linde several months to correct it, since anything he sent to the West would have to be passed by Soviet censorship, which was neither very skillful nor very quick with scientific papers. Instead, I wrote a short paper with Ian Moss in the same journal in which we pointed out this problem with the bubble and showed how it could be resolved. The day after I got back from Moscow I set out for Philadelphia, where I was due to receive a medal from the Franklin Institute. My secretary, Judy Fella, had used her not inconsiderable charm to persuade British Airways to give herself and me free seats on a Concorde as a publicity venture. However, I .was held up on my way to the airport by heavy rain and I missed the plane. Nevertheless, I got to Philadelphia in the end and received my medal. I was then asked to give a seminar on the inflationary universe at Drexel University in Philadelphia. I gave the same seminar about the problems of the inflationary universe, just as in Moscow. A very similar idea to Linde’s was put forth independently a few months later by Paul Steinhardt and Andreas Albrecht of the University of Pennsylvania. They are now given joint credit with Linde for what is called “the new inflationary model,” based on the idea of a slow breaking of symmetry. (The old inflationary model was Guth’s original suggestion of fast symmetry breaking with the formation of bubbles.) The new inflationary model was a good attempt to explain why the universe is the way it is. However, I and several other people showed that, at least in its original form, it predicted much greater variations in the temperature of the microwave background radiation than are observed. Later work has also cast doubt on whether there could be a phase transition in the very early universe of the kind required. In my personal opinion, the new inflationary model is now dead as a scientific theory, although a lot of people do not seem to have heard of its demise and are still writing papers as if it were viable. A better model, called the chaotic inflationary model, was put forward by Linde in 1983. In this there is no phase transition or supercooling. Instead, there is a spin 0 field, which, because of quantum fluctuations, would have large values in some regions of the early universe. The energy of the field in those regions would behave like a cosmological constant. It would have a repulsive gravitational effect, and thus make those regions expand in an inflationary manner. As they expanded, the energy of the field in them would slowly decrease until the inflationary expansion changed to an expansion like that in the hot big bang model. One of these regions would become what we now see as the observable universe. This model has all the advantages of the earlier inflationary models, but it does not depend on a dubious phase transition, and it can moreover give a reasonable size for the fluctuations in the temperature of the microwave background that agrees with observation. This work on inflationary models showed that the present state of the universe could have arisen from quite a large number of different initial configurations. This is important, because it shows that the initial state of the part of the universe that we inhabit did not have to be chosen with great care. So we may, if we wish, use the weak anthropic principle to explain why the universe looks the way it does now. It cannot be the case, however, that every initial configuration would have led to a universe like the one we observe. One can show this by considering a very different state for the universe at the present time, say, a very lumpy and irregular one. One could use the laws of science to evolve the universe back in time to determine its configuration at earlier times. According to the singularity theorems of classical general relativity, there would still have been a big bang singularity. If you evolve such a universe forward in time according to the laws of science, you will end up with the lumpy and irregular state you started with. Thus there must have been initial configurations that would not have given rise to a universe like the one we see today. So even the inflationary model does not tell us why the initial configuration was not such as to produce something very different from what we observe. Must we turn to the anthropic principle for an explanation? Was it all just a lucky chance? That would seem a counsel of despair, a negation of all our hopes of understanding the underlying order of the universe. In order to predict how the universe should have started off, one needs laws that hold at the beginning of time. If the classical theory of general relativity was correct, the singularity theorems that Roger Penrose and I proved show that file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (7 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 the beginning of time would have been a point of infinite density and infinite curvature of space-time. All the known laws of science would break down at such a point. One might suppose that there were new laws that held at singularities, but it would be very difficult even to formulate such laws at such badly behaved points, and we would have no guide from observations as to what those laws might be. However, what the singularity theorems really indicate is that the gravitational field becomes so strong that quantum gravitational effects become important: classical theory is no longer a good description of the universe. So one has to use a quantum theory of gravity to discuss the very early stages of the universe. As we shall see, it is possible in the quantum theory for the ordinary laws of science to hold everywhere, including at the beginning of time: it is not necessary to postulate new laws for singularities, because there need not be any singularities in the quantum theory. We don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. However, we are fairly certain of some features that such a unified theory should have. One is that it should incorporate Feynman’s proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle does not have just a single history, as it would in a classical theory. Instead, it is supposed to follow every possible path in space-time, and with each of these histories there are associated a couple of numbers, one represent-ing the size of a wave and the other representing its position in the cycle (its phase). The probability that the particle, say, passes through some particular point is found by adding up the waves associated with every possible history that passes through that point. When one actually tries to perform these sums, however, one runs into severe technical problems. The only way around these is the following peculiar prescription: one must add up the waves for particle histories that are not in the “real” time that you and I experience but take place in what is called imaginary time. Imaginary time may sound like science fiction but it is in fact a well-defined mathematical concept. If we take any ordinary (or “real”) number and multiply it by itself, the result is a positive number. (For example, 2 times 2 is 4, but so is – 2 times – 2.) There are, however, special numbers (called imaginary numbers) that give negative numbers when multiplied by themselves. (The one called i, when multiplied by itself, gives – 1, 2i multiplied by itself gives – 4, and so on.) One can picture real and imaginary numbers in the following way: The real numbers can be represented by a line going from left to right, with zero in the middle, negative numbers like – 1, – 2, etc. on the left, and positive numbers, 1, 2, etc. on the right. Then imaginary numbers are represented by a line going up and down the page, with i, 2i, etc. above the middle, and – i, – 2i, etc. below. Thus imaginary numbers are in a sense numbers at right angles to ordinary real numbers. To avoid the technical difficulties with Feynman’s sum over histories, one must use imaginary time. That is to say, for the purposes of the calculation one must measure time using imaginary numbers, rather than real ones. This has an interesting effect on space-time: the distinction between time and space disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be Euclidean, after the ancient Greek Euclid, who founded the study of the geometry of two-dimensional surfaces. What we now call Euclidean space-time is very similar except that it has four dimensions instead of two. In Euclidean space-time there is no difference between the time direction and directions in space. On the other hand, in real space-time, in which events are labeled by ordinary, real values of the time coordinate, it is easy to tell the difference – the time direction at all points lies within the light cone, and space directions lie outside. In any case, as far as everyday quantum mechanics is concerned, we may regard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculate answers about real space-time. A second feature that we believe must be part of any ultimate theory is Einstein’s idea that the gravitational field is represented by curved space-time: particles try to follow the nearest thing to a straight path in a curved space, but because space-time is not flat their paths appear to be bent, as if by a gravitational field. When we apply Feynman’s sum over histories to Einstein’s view of gravity, the analogue of the history of a particle is now a complete curved space-time that represents the history of the whole universe. To avoid the technical difficulties in actually performing the sum over histories, these curved space-times must be taken to be Euclidean. That is, time is imaginary and is indistinguishable from directions in space. To calculate the probability of finding a real space-time with some certain property, such as looking the same at every point and in every direction, one adds up the waves associated with all the histories that have that property. In the classical theory of general relativity, there are many different possible curved space-times, each corresponding to a different initial state of the universe. If we knew the initial state of our universe, we would know its entire history. Similarly, in the quantum theory of gravity, there are many different possible quantum states for the universe. Again, if we knew how the Euclidean curved space-times in the sum over histories behaved at early times, we would know the quantum state of the universe. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (8 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 In the classical theory of gravity, which is based on real space-time, there are only two possible ways the universe can behave: either it has existed for an infinite time, or else it had a beginning at a singularity at some finite time in the past. In the quantum theory of gravity, on the other hand, a third possibility arises. Because one is using Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would be like the surface of the earth, only with two more dimensions. The surface of the earth is finite in extent but it doesn’t have a boundary or edge: if you sail off into the sunset, you don’t fall off the edge or run into a singularity. (I know, because I have been round the world!) If Euclidean space-time stretches back to infinite imaginary time, or else starts at a singularity in imaginary time, we have the same problem as in the classical theory of specifying the initial state of the universe: God may know how the universe began, but we cannot give any particular reason for thinking it began one way rather than another. On the other hand, the quantum theory of gravity has opened up a new possibility, in which there would be no boundary to space-time and so there would be no need to specify the behavior at the boundary. There would be no singularities at which the laws of science broke down, and no edge of space-time at which one would have to appeal to God or some new law to set the boundary conditions for space-time. One could say: “The boundary condition of the universe is that it has no boundary.” The universe would be completely self-contained and not affected by anything outside itself. It would neither be created nor destroyed, It would just BE. It was at the conference in the Vatican mentioned earlier that I first put forward the suggestion that maybe time and space together formed a surface that was finite in size but did not have any boundary or edge. My paper was rather mathematical, however, so its implications for the role of God in the creation of the universe were not generally recognized at the time (just as well for me). At the time of the Vatican conference, I did not know how to use the “no boundary” idea to make predictions about the universe. However, I spent the following sum-mer at the University of California, Santa Barbara. There a friend and colleague of mine, Jim Hartle, worked out with me what conditions the universe must satisfy if space-time had no boundary. When I returned to Cambridge, I continued this work with two of my research students, Julian Luttrel and Jonathan Halliwell. I’d like to emphasize that this idea that time and space should be finite “without boundary” is just a proposal: it cannot be deduced from some other principle. Like any other scientific theory, it may initially be put forward for aesthetic or metaphysical reasons, but the real test is whether it makes predictions that agree with observation. This, how-ever, is difficult to determine in the case of quantum gravity, for two reasons. First, as will be explained in Chapter 11, we are not yet sure exactly which theory successfully combines general relativity and quantum mechanics, though we know quite a lot about the form such a theory must have. Second, any model that described the whole universe in detail would be much too complicated mathematically for us to be able to calculate exact predictions. One therefore has to make simplifying assumptions and approximations – and even then, the problem of extracting predictions remains a formidable one. Each history in the sum over histories will describe not only the space-time but everything in it as well, including any complicated organisms like human beings who can observe the history of the universe. This may provide another justification for the anthropic principle, for if all the histories are possible, then so long as we exist in one of the histories, we may use the anthropic principle to explain why the universe is found to be the way it is. Exactly what meaning can be attached to the other histories, in which we do not exist, is not clear. This view of a quantum theory of gravity would be much more satisfactory, however, if one could show that, using the sum over histories, our universe is not just one of the possible histories but one of the most probable ones. To do this, we must perform the sum over histories for all possible Euclidean space-times that have no boundary. Under the “no boundary” proposal one learns that the chance of the universe being found to be following most of the possible histories is negligible, but there is a particular family of histories that are much more probable than the others. These histories may be pictured as being like the surface of the earth, with the distance from the North Pole representing imaginary time and the size of a circle of constant distance from the North Pole representing the spatial size of the universe. The universe starts at the North Pole as a single point. As one moves south, the circles of latitude at constant distance from the North Pole get bigger, corresponding to the universe expanding with imaginary time Figure 8:1. The universe would reach a maximum size at the equator and would contract with increasing imaginary time to a single point at the South Pole. Ever though the universe would have zero size at the North and South Poles, these points would not be singularities, any more than the North aid South Poles on the earth are singular. The laws of science will hold at them, just as they do at the North and South Poles on the earth. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (9 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 Figure 8:1 The history of the universe in real time, however, would look very different. At about ten or twenty thousand million years ago, it would have a minimum size, which was equal to the maximum radius of the history in imaginary time. At later real times, the universe would expand like the chaotic inflationary model proposed by Linde (but one would not now have to assume that the universe was created somehow in the right sort of state). The universe would expand to a very large size Figure 8:1 and eventually it would collapse again into what looks like a singularity in real time. Thus, in a sense, we are still all doomed, even if we keep away from black holes. Only if we could picture the universe in terms of imaginary time would there be no singularities. If the universe really is in such a quantum state, there would be no singularities in the history of the universe in imaginary time. It might seem therefore that my more recent work had completely undone the results of my earlier work on singularities. But, as indicated above, the real importance of the singularity theorems was that they showed that the gravitational field must become so strong that quantum gravitational effects could not be ignored. This in turn led to the idea that the universe could be finite in imaginary time but without boundaries or singularities. When one goes back to the real time in which we live, however, there will still appear to be singularities. The poor astronaut who falls into a black hole will still come to a sticky end; only if he lived in imaginary time would he encounter no singularities. This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a figment of our imaginations. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just an idea that we invent to help us describe what we think the universe is like. But according to the approach I described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter of which is the more useful description. One can also use the sum over histories, along with the no boundary proposal, to find which properties of the universe are likely to occur together. For example, one can calculate the probability that the universe is expanding at nearly the same rate in all different directions at a time when the density of the universe has its present value. In the simplified models that have been examined so far, this probability turns out to be high; that is, the proposed no boundary condition leads to the prediction that it is extremely probable that the present rate of expansion of the universe is almost the same in each direction. This is consistent with the observations of the microwave background radiation, which show that it has almost exactly the same intensity in any direction. If the universe were expanding faster in some directions than in others, the intensity of the radiation in those directions would be reduced by an file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (10 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 8 additional red shift. Further predictions of the no boundary condition are currently being worked out. A particularly interesting problem is the size of the small departures from uniform density in the early universe that caused the formation first of the galaxies, then of stars, and finally of us. The uncertainty principle implies that the early universe cannot have been completely uniform because there must have been some uncertainties or fluctuations in the positions and velocities of the particles. Using the no boundary condition, we find that the universe must in fact have started off with just the minimum possible non-uniformity allowed by the uncertainty principle. The universe would have then undergone a period of rapid expansion, as in the inflationary models. During this period, the initial non-uniformities would have been amplified until they were big enough to explain the origin of the structures we observe around us. In 1992 the Cosmic Background Explorer satellite (COBE) first detected very slight variations in the intensity of the microwave background with direction. The way these non-uniformities depend on direction seems to agree with the predictions of the inflationary model and the no boundary proposal. Thus the no boundary proposal is a good scientific theory in the sense of Karl Popper: it could have been falsified by observations but instead its predictions have been confirmed. In an expanding universe in which the density of matter varied slightly from place to place, gravity would have caused the denser regions to slow down their expansion and start contracting. This would lead to the formation of galaxies, stars, and eventually even insignificant creatures like ourselves. Thus all the complicated structures that we see in the universe might be explained by the no boundary condition for the universe together with the uncertainty principle of quantum mechanics. The idea that space and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people have come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws. However, the laws do not tell us what the universe should have looked like when it started – it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator? PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/g.html (11 of 11) [2/20/2001 3:15:29 AM]
A Brief History of Time - Stephen Hawking... Chapter 9 CHAPTER 9 THE ARROW OF TIME In previous chapters we have seen how our views of the nature of time have changed over the years. Up to the beginning of this century people believed in an absolute time. That is, each event could be labeled by a number called “time” in a unique way, and all good clocks would agree on the time interval between two events. However, the discovery that the speed of light appeared the same to every observer, no matter how he was moving, led to the theory of relativity – and in that one had to abandon the idea that there was a unique absolute time. Instead, each observer would have his own measure of time as recorded by a clock that he carried: clocks carried by different observers would not necessarily agree. Thus time became a more personal concept, relative to the observer who measured it. When one tried to unify gravity with quantum mechanics, one had to introduce the idea of “imaginary” time. Imaginary time is indistinguishable from directions in space. If one can go north, one can turn around and head south; equally, if one can go forward in imaginary time, one ought to be able to turn round and go backward. This means that there can be no important difference between the forward and backward directions of imaginary time. On the other hand, when one looks at “real” time, there’s a very big difference between the forward and backward directions, as we all know. Where does this difference between the past and the future come from? Why do we remember the past but not the future? The laws of science do not distinguish between the past and the future. More precisely, as explained earlier, the laws of science are unchanged under the combination of operations (or symmetries) known as C, P, and T. (C means changing particles for antiparticles. P means taking the mirror image, so left and right are interchanged. And T means reversing the direction of motion of all particles: in effect, running the motion backward.) The laws of science that govern the behavior of matter under all normal situations are unchanged under the combination of the two operations C and P on their own. In other words, life would be just the same for the inhabitants of another planet who were both mirror images of us and who were made of antimatter, rather than matter. If the laws of science are unchanged by the combination of operations C and P, and also by the combination C, P, and T, they must also be unchanged under the operation T alone. Yet there is a big difference between the forward and backward directions of real time in ordinary life. Imagine a cup of water falling off a table and breaking into pieces on the floor. If you take a film of this, you can easily tell whether it is being run forward or backward. If you run it backward you will see the pieces suddenly gather themselves together off the floor and jump back to form a whole cup on the table. You can tell that the film is being run backward because this kind of behavior is never observed in ordinary life. If it were, crockery manufacturers would go out of business. The explanation that is usually given as to why we don’t see broken cups gathering themselves together off the floor and jumping back onto the table is that it is forbidden by the second law of thermodynamics. This says that in any closed system disorder, or entropy, always increases with time. In other words, it is a form of Murphy’s law: things always tend to go wrong! An intact cup on the table is a state of high order, but a broken cup on the floor is a disordered state. One can go readily from the cup on the table in the past to the broken cup on the floor in the future, but not the other way round. The increase of disorder or entropy with time is one example of what is called an arrow of time, something that distinguishes the past from the future, giving a direction to time. There are at least three different arrows of time. First, there is the thermodynamic arrow of time, the direction of time in which disorder or entropy increases. Then, there is the psychological arrow of time. This is the direction in which we feel time passes, the direction in which we remember the past but not the future. Finally, there is the cosmological arrow of time. This is the direction of time in which the universe is expanding rather than contracting. In this chapter I shall argue that the no boundary condition for the universe, together with the weak anthropic principle, can explain why all three arrows point in the same direction – and moreover, why a well-defined arrow of time should exist at all. I shall argue that the psychological arrow is determined by the thermodynamic arrow, file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/h.html (1 of 5) [2/20/2001 3:15:38 AM]
A Brief History of Time - Stephen Hawking... Chapter 9 and that these two arrows necessarily always point in the same direction. If one assumes the no boundary condition for the universe, we shall see that there must be well-defined thermodynamic and cosmological arrows of time, but they will not point in the same direction for the whole history of the universe. However, I shall argue that it is only when they do point in the same direction that conditions are suitable for the development of intelligent beings who can ask the question: why does disorder increase in the same direction of time as that in which the universe expands? I shall discuss first the thermodynamic arrow of time. The second law of thermodynamics results from the fact that there are always many more disordered states than there are ordered ones. For example, consider the pieces of a jigsaw in a box. There is one, and. only one, arrangement in which the pieces make a complete picture. On the other hand, there are a very large number of arrangements in which the pieces are disordered and don’t make a picture. Suppose a system starts out in one of the small number of ordered states. As time goes by, the system will evolve according to the laws of science and its state will change. At a later time, it is more probable that the system will be in a disordered state than in an ordered one because there are more disordered states. Thus disorder will tend to increase with time if the system obeys an initial condition of high order. Suppose the pieces of the jigsaw start off in a box in the ordered arrangement in which they form a picture. If you shake the box, the pieces will take up another arrangement. This will probably be a disordered arrangement in which the pieces don’t form a proper picture, simply because there are so many more disordered arrangements. Some groups of pieces may still form parts of the picture, but the more you shake the box, the more likely it is that these groups will get broken up and the pieces will be in a completely jumbled state in which they don’t form any sort of picture. So the disorder of the pieces will probably increase with time if the pieces obey the initial condition that they start off in a condition of high order. Suppose, however, that God decided that the universe should finish up in a state of high order but that it didn’t matter what state it started in. At early times the universe would probably be in a disordered state. This would mean that disorder would decrease with time. You would see broken cups gathering themselves together and jumping back onto the table. However, any human beings who were observing the cups would be living in a universe in which disorder decreased with time. I shall argue that such beings would have a psychological arrow of time that was backward. That is, they would remember events in the future, and not remember events in their past. When the cup was broken, they would remember it being on the table, but when it was on the table, they would not remember it being on the floor. It is rather difficult to talk about human memory because we don’t know how the brain works in detail. We do, however, know all about how computer memories work. I shall therefore discuss the psychological arrow of time for computers. I think it is reasonable to assume that the arrow for computers is the same as that for humans. If it were not, one could make a killing on the stock exchange by having a computer that would remember tomorrow’s prices! A computer memory is basically a device containing elements that can exist in either of two states. A simple example is an abacus. In its simplest form, this consists of a number of wires; on each wire there are a number of beads that can be put in one of two positions. Before an item is recorded in a computer’s memory, the memory is in a disordered state, with equal probabilities for the two possible states. (The abacus beads are scattered randomly on the wires of the abacus.) After the memory interacts with the system to be remembered, it will definitely be in one state or the other, according to the state of the system. (Each abacus bead will be at either the left or the right of the abacus wire.) So the memory has passed from a disordered state to an ordered one. However, in order to make sure that the memory is in the right state, it is necessary to use a certain amount of energy (to move the bead or to power the computer, for example). This energy is dissipated as heat, and increases the amount of disorder in the universe. One can show that this increase in disorder is always greater than the increase in the order of the memory itself. Thus the heat expelled by the computer’s cooling fan means that when a computer records an item in memory, the total amount of disorder in the universe still goes up. The direction of time in which a computer remembers the past is the same as that in which disorder increases. Our subjective sense of the direction of time, the psychological arrow of time, is therefore determined within our brain by the thermodynamic arrow of time. Just like a computer, we must remember things in the order in which entropy increases. This makes the second law of thermodynamics almost trivial. Disorder increases with time file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/h.html (2 of 5) [2/20/2001 3:15:38 AM]
A Brief History of Time - Stephen Hawking... Chapter 9 because we measure time in the direction in which disorder increases You can’t have a safer bet than that! But why should the thermodynamic arrow of time exist at all? Or, in other words, why should the universe be in a state of high order at one end of time, the end that we call the past? Why is it not in a state of complete disorder at all times? After all, this might seem more probable. And why is the direction of time in which disorder increases the same as that in which the universe expands? In the classical theory of general relativity one cannot predict how the universe would have begun because all the known laws of science would have broken down at the big bang singularity. The universe could have started out in a very smooth and ordered state. This would have led to well-defined thermodynamic and cosmological arrows of time, as we observe. But it could equally well have started out in a very lumpy and disordered state. In that case, the universe would already be in a state of complete disorder, so disorder could not increase with time. It would either stay constant, in which case there would be no well-defined thermodynamic arrow of time, or it would decrease, in which case the thermodynamic arrow of time would point in the opposite direction to the cosmological arrow. Neither of these possibilities agrees with what we observe. However, as we have seen, classical general relativity predicts its own downfall. When the curvature of space-time becomes large, quantum gravitational effects will become important and the classical theory will cease to be a good description of the universe. One has to use a quantum theory of gravity to understand how the universe began. In a quantum theory of gravity, as we saw in the last chapter, in order to specify the state of the universe one would still have to say how the possible histories of the universe would behave at the boundary of space-time in the past. One could avoid this difficulty of having to describe what we do not and cannot know only if the histories satisfy the no boundary condition: they are finite in extent but have no boundaries, edges, or singularities. In that case, the beginning of time would be a regular, smooth point of space-time and the universe would have begun its expansion in a very smooth and ordered state. It could not have been completely uniform, because that would violate the uncertainty principle of quantum theory. There had to be small fluctuations in the density and velocities of particles. The no boundary condition, however, implied that these fluctuations were as small as they could be, consistent with the uncertainty principle. The universe would have started off with a period of exponential or “inflationary” expansion in which it would have increased its size by a very large factor. During this expansion, the density fluctuations would have remained small at first, but later would have started to grow. Regions in which the density was slightly higher than average would have had their expansion slowed down by the gravitational attraction of the extra mass. Eventually, such regions would stop expanding and collapse to form galaxies, stars, and beings like us. The universe would have started in a smooth and ordered state, and would become lumpy and disordered as time went on. This would explain the existence of the thermodynamic arrow of time. But what would happen if and when the universe stopped expanding and began to contract? Would the thermodynamic arrow reverse and disorder begin to decrease with time? This would lead to all sorts of science-fiction-like possibilities for people who survived from the expanding to the contracting phase. Would they see broken cups gathering themselves together off the floor and jumping back onto the table? Would they be able to remember tomorrow’s prices and make a fortune on the stock market? It might seem a bit academic to worry about what will happen when the universe collapses again, as it will not start to contract for at least another ten thousand million years. But there is a quicker way to find out what will happen: jump into a black hole. The collapse of a star to form a black hole is rather like the later stages of the collapse of the whole universe. So if disorder were to decrease in the contracting phase of the universe, one might also expect it to decrease inside a black hole. So perhaps an astronaut who fell into a black hole would be able to make money at roulette by remembering where the ball went before he placed his bet. (Unfortunately, however, he would not have long to play before he was turned to spaghetti. Nor would he be able to let us know about the reversal of the thermodynamic arrow, or even bank his winnings, because he would be trapped behind the event horizon of the black hole.) At first, I believed that disorder would decrease when the universe recollapsed. This was because I thought that the universe had to return to a smooth and ordered state when it became small again. This would mean that the contracting phase would be like the time reverse of the expanding phase. People in the contracting phase would live their lives backward: they would die before they were born and get younger as the universe file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/h.html (3 of 5) [2/20/2001 3:15:38 AM]
A Brief History of Time - Stephen Hawking... Chapter 9 contracted. This idea is attractive because it would mean a nice symmetry between the expanding and contracting phases. However, one cannot adopt it on its own, independent of other ideas about the universe. The question is: is it implied by the no boundary condition, or is it inconsistent with that condition? As I said, I thought at first that the no boundary condition did indeed imply that disorder would decrease in the contracting phase. I was misled partly by the analogy with the surface of the earth. If one took the beginning of the universe to correspond to the North Pole, then the end of the universe should be similar to the beginning, just as the South Pole is similar to the North. However, the North and South Poles correspond to the beginning and end of the universe in imaginary time. The beginning and end in real time can be very different from each other. I was also misled by work I had done on a simple model of the universe in which the collapsing phase looked like the time reverse of the expanding phase. However, a colleague of mine, Don Page, of Penn State University, pointed out that the no boundary condition did not require the contracting phase necessarily to be the time reverse of the expanding phase. Further, one of my students, Raymond Laflamme, found that in a slightly more complicated model, the collapse of the universe was very different from the expansion. I realized that I had made a mistake: the no boundary condition implied that disorder would in fact continue to increase during the contraction. The thermodynamic and psychological arrows of time would not reverse when the universe begins to recontract, or inside black holes. What should you do when you find you have made a mistake like that? Some people never admit that they are wrong and continue to find new, and often mutually inconsistent, arguments to support their case – as Eddington did in opposing black hole theory. Others claim to have never really supported the incorrect view in the first place or, if they did, it was only to show that it was inconsistent. It seems to me much better and less confusing if you admit in print that you were wrong. A good example of this was Einstein, who called the cosmological constant, which he introduced when he was trying to make a static model of the universe, the biggest mistake of his life. To return to the arrow of time, there remains the question: why do we observe that the thermodynamic and cosmological arrows point in the same direction? Or in other words, why does disorder increase in the same direction of time as that in which the universe expands? If one believes that the universe will expand and then contract again, as the no boundary proposal seems to imply, this becomes a question of why we should be in the expanding phase rather than the contracting phase. One can answer this on the basis of the weak anthropic principle. Conditions in the contracting phase would not be suitable for the existence of intelligent beings who could ask the question: why is disorder increasing in the same direction of time as that in which the universe is expanding? The inflation in the early stages of the universe, which the no boundary proposal predicts, means that the universe must be expanding at very close to the critical rate at which it would just avoid recollapse, and so will not recollapse for a very long time. By then all the stars will have burned out and the protons and neutrons in them will probably have decayed into light particles and radiation. The universe would be in a state of almost complete disorder. There would be no strong thermodynamic arrow of time. Disorder couldn’t increase much because the universe would be in a state of almost complete disorder already. However, a strong thermodynamic arrow is necessary for intelligent life to operate. In order to survive, human beings have to consume food, which is an ordered form of energy, and convert it into heat, which is a disordered form of energy. Thus intelligent life could not exist in the contracting phase of the universe. This is the explanation of why we observe that the thermodynamic and cosmological arrows of time point in the same direction. It is not that the expansion of the universe causes disorder to increase. Rather, it is that the no boundary condition causes disorder to increase and the conditions to be suitable for intelligent life only in the expanding phase. To summarize, the laws of science do not distinguish between the forward and backward directions of time. However, there are at least three arrows of time that do distinguish the past from the future. They are the thermodynamic arrow, the direction of time in which disorder increases; the psychological arrow, the direction of time in which we remember the past and not the future; and the cosmological arrow, the direction of time in which the universe expands rather than contracts. I have shown that the psychological arrow is essentially the same as the thermodynamic arrow, so that the two would always point in the same direction. The no boundary proposal for the universe predicts the existence of a well-defined thermodynamic arrow of time because the universe must start off in a smooth and ordered state. And the reason we observe this thermodynamic arrow to file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/h.html (4 of 5) [2/20/2001 3:15:38 AM]
A Brief History of Time - Stephen Hawking... Chapter 9 agree with the cosmological arrow is that intelligent beings can exist only in the expanding phase. The contracting phase will be unsuitable because it has no strong thermodynamic arrow of time. The progress of the human race in understanding the universe has established a small corner of order in an increasingly disordered universe. If you remember every word in this book, your memory will have recorded about two million pieces of information: the order in your brain will have increased by about two million units. However, while you have been reading the book, you will have converted at least a thousand calories of ordered energy, in the form of food, into disordered energy, in the form of heat that you lose to the air around you by convection and sweat. This will increase the disorder of the universe by about twenty million million million million units – or about ten million million million times the increase in order in your brain – and that’s if you remember everything in this book. In the next chapter but one I will try to increase the order in our neck of the woods a little further by explaining how people are trying to fit together the partial theories I have described to form a complete unified theory that would cover everything in the universe. PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/h.html (5 of 5) [2/20/2001 3:15:38 AM]
A Brief History of Time - Stephen Hawking... Chapter 10 CHAPTER 10 WORMHOLES AND TIME TRAVEL The last chapter discussed why we see time go forward: why disorder increases and why we remember the past but not the future. Time was treated as if it were a straight railway line on which one could only go one way or the other. But what if the railway line had loops and branches so that a train could keep going forward but come back to a station it had already passed? In other words, might it be possible for someone to travel into the future or the past? H. G. Wells in The Time Machine explored these possibilities as have countless other writers of science fiction. Yet many of the ideas of science fiction, like submarines and travel to the moon, have become matters of science fact. So what are the prospects for time travel? The first indication that the laws of physics might really allow people to travel in time came in 1949 when Kurt Godel discovered a new space-time allowed by general relativity. Godel was a mathematician who was famous for proving that it is impossible to prove all true statements, even if you limit yourself to trying to prove all the true statements in a subject as apparently cut and dried as arithmetic. Like the uncertainty principle, Godel’s incompleteness theorem may be a fundamental limitation on our ability to understand and predict the universe, but so far at least it hasn’t seemed to be an obstacle in our search for a complete unified theory. Godel got to know about general relativity when he and Einstein spent their later years at the Institute for Advanced Study in Princeton. His space-time had the curious property that the whole universe was rotating. One might ask: “Rotating with respect to what?” The answer is that distant matter would be rotating with respect to directions that little tops or gyroscopes point in. This had the side effect that it would be possible for someone to go off in a rocket ship and return to earth before he set out. This property really upset Einstein, who had thought that general relativity wouldn’t allow time travel. However, given Einstein’s record of ill-founded opposition to gravitational collapse and the uncertainty principle, maybe this was an encouraging sign. The solution Godel found doesn’t correspond to the universe we live in because we can show that the universe is not rotating. It also had a non-zero value of the cosmological constant that Einstein introduced when he thought the universe was unchanging. After Hubble discovered the expansion of the universe, there was no need for a cosmological constant and it is now generally believed to be zero. However, other more reasonable space-times that are allowed by general relativity and which permit travel into the past have since been found. One is in the interior of a rotating black hole. Another is a space-time that contains two cosmic strings moving past each other at high speed. As their name suggests, cosmic strings are objects that are like string in that they have length but a tiny cross section. Actually, they are more like rubber bands because they are under enormous tension, something like a million million million million tons. A cosmic string attached to the earth could accelerate it from 0 to 60 mph in 1/30th of a second. Cosmic strings may sound like pure science fiction but there are reasons to believe they could have formed in the early universe as a result of symmetry-breaking of the kind discussed in Chapter 5. Because they would be under enormous tension and could start in any configuration, they might accelerate to very high speeds when they straighten out. The Godel solution and the cosmic string space-time start out so distorted that travel into the past was always possible. God might have created such a warped universe but we have no reason to believe he did. Observations of the microwave background and of the abundances of the light elements indicate that the early universe did not have the kind of curvature required to allow time travel. The same conclusion follows on theoretical grounds if the no boundary proposal is correct. So the question is: if the universe starts out without the kind of curvature required for time travel, can we subsequently warp local regions of space-time sufficiently to allow it? A closely related problem that is also of concern to writers of science fiction is rapid interstellar or intergalactic file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/i.html (1 of 5) [2/20/2001 3:15:45 AM]
A Brief History of Time - Stephen Hawking... Chapter 10 travel. According to relativity, nothing can travel faster than light. If we therefore sent a spaceship to our nearest neighboring star, Alpha Centauri, which is about four light-years away, it would take at least eight years before we could expect the travelers to return and tell us what they had found. If the expedition were to the center of our galaxy, it would be at least a hundred thousand years before it came back. The theory of relativity does allow one consolation. This is the so-called twins paradox mentioned in Chapter 2. Because there is no unique standard of time, but rather observers each have their own time as measured by clocks that they carry with them, it is possible for the journey to seem to be much shorter for the space travelers than for those who remain on earth. But there would not be much joy in returning from a space voyage a few years older to find that everyone you had left behind was dead and gone thousands of years ago. So in order to have any human interest in their stories, science fiction writers had to suppose that we would one day discover how to travel faster than light. What most of thee authors don’t seem to have realized is that if you can travel faster than light, the theory of relativity implies you can also travel back in the, as the following limerick says: There was a young lady of Wight Who traveled much faster than light. She departed one day, In a relative way, And arrived on the previous night The point is that the theory of relativity says hat there is no unique measure of time that all observers will agree on Rather, each observer has his or her own measure of time. If it is possible for a rocket traveling below the speed of light to get from event A (say, the final of the 100-meter race of the Olympic Games in 202) to event B (say, the opening of the 100,004th meeting of the Congress of Alpha Centauri), then all observers will agree that event A happened before event B according to their times. Suppose, however, that the spaceship would have to travel faster than light to carry the news of the race to the Congress. Then observers moving at different speeds can disagree about whether event A occurred before B or vice versa. According to the time of an observer who is at rest with respect to the earth, it may be that the Congress opened after the race. Thus this observer would think that a spaceship could get from A to B in time if only it could ignore the speed-of-light speed limit. However, to an observer at Alpha Centauri moving away from the earth at nearly the speed of light, it would appear that event B, the opening of the Congress, would occur before event A, the 100-meter race. The theory of relativity says that the laws of physics appear the same to observers moving at different speeds. This has been well tested by experiment and is likely to remain a feature even if we find a more advanced theory to replace relativity Thus the moving observer would say that if faster-than-light travel is possible, it should be possible to get from event B, the opening of the Congress, to event A, the 100-meter race. If one went slightly faster, one could even get back before the race and place a bet on it in the sure knowledge that one would win. There is a problem with breaking the speed-of-light barrier. The theory of relativity says that the rocket power needed to accelerate a spaceship gets greater and greater the nearer it gets to the speed of light. We have experimental evidence for this, not with spaceships but with elementary particles in particle accelerators like those at Fermilab or CERN (European Centre for Nuclear Research). We can accelerate particles to 99.99 percent of the speed of light, but however much power we feed in, we can’t get them beyond the speed-of-light barrier. Similarly with spaceships: no matter how much rocket power they have, they can’t accelerate beyond the speed of light. That might seem to rule out both rapid space travel and travel back in time. However, there is a possible way out. It might be that one could warp space-time so that there was a shortcut between A and B One way of doing this would be to create a wormhole between A and B. As its name suggests, a wormhole is a thin tube of space-time which can connect two nearly flat regions far apart. There need be no relation between the distance through the wormhole and the separation of its ends in the nearly Hat background. Thus one could imagine that one could create or find a wormhole that world lead from the vicinity of the Solar System to Alpha Centauri. The distance through the wormhole might be only a few million miles even though earth and Alpha Centauri are twenty million million miles apart in ordinary space. This would allow news of the 100-meter race to reach the opening of the Congress. But then an observer moving toward 6e earth should also be able to find another wormhole that would enable him to get from the opening of file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/i.html (2 of 5) [2/20/2001 3:15:45 AM]
A Brief History of Time - Stephen Hawking... Chapter 10 the Congress on Alpha Centauri back to earth before the start of the race. So wormholes, like any other possible form of travel faster than light, would allow one to travel into the past. The idea of wormholes between different regions of space-time was not an invention of science fiction writers but came from a very respectable source. In 1935, Einstein and Nathan Rosen wrote a paper in which they showed that general relativity allowed what they called “bridges,” but which are now known as wormholes. The Einstein-Rosen bridges didn’t last long enough for a spaceship to get through: the ship would run into a singularity as the wormhole pinched off. However, it has been suggested that it might be possible for an advanced civilization to keep a wormhole open. To do this, or to warp space-time in any other way so as to permit time travel, one can show that one needs a region of space-time with negative curvature, like the surface of a saddle. Ordi-nary matter, which has a positive energy density, gives space-time a positive curvature, like the surface of a sphere. So what one needs, in order to warp space-time in a way that will allow travel into the past, is matter with negative energy density. Energy is a bit like money: if you have a positive balance, you can distribute it in various ways, but according to the classical laws that were believed at the beginning of the century, you weren’t allowed to be overdrawn. So these classical laws would have ruled out any possibility of time travel. However, as has been described in earlier chapters, the classical laws were superseded by quantum laws based on the uncertainty principle. The quantum laws are more liberal and allow you to be overdrawn on one or two accounts provided the total balance is positive. In other words, quantum theory allows the energy density to be negative in some places, provided that this is made up for by positive energy densities in other places, so that the total energy re-mains positive. An example of how quantum theory can allow negative energy densities is provided by what is called the Casimir effect. As we saw in Chapter 7, even what we think of as “empty” space is filled with pairs of virtual particles and antiparticles that appear together, move apart, and come back together and annihilate each other. Now, suppose one has two parallel metal plates a short distance apart. The plates will act like mirrors for the virtual photons or particles of light. In fact they will form a cavity between them, a bit like an organ pipe that will resonate only at certain notes. This means that virtual photons can occur in the space between the plates only if their wavelengths (the distance between the crest of one wave and the next) fit a whole number of times into the gap between the plates. If the width of a cavity is a whole number of wavelengths plus a fraction of a wave-length, then after some reflections backward and forward between the plates, the crests of one wave will coincide with the troughs of another and the waves will cancel out. Because the virtual photons between the plates can have only the resonant wavelengths, there will be slightly fewer of them than in the region outside the plates where virtual photons can have any wavelength. Thus there will be slightly fewer virtual photons hitting the inside surfaces of the plates than the outside surfaces. One would therefore expect a force on the plates, pushing them toward each other. This force has actually been detected and has the predicted value. Thus we have experimental evidence that virtual particles exist and have real effects. The fact that there are fewer virtual photons between the plates means that their energy density will be less than elsewhere. But the total energy density in “empty” space far away from the plates must be zero, because otherwise the energy density would warp the space and it would not be almost flat. So, if the energy density between the plates is less than the energy density far away, it must be negative. We thus have experimental evidence both that space-time can be warped (from the bending of light during eclipses) and that it can be curved in the way necessary to allow time travel (from the Casimir effect). One might hope therefore that as we advance in science and technology, we would eventually manage to build a time machine. But if so, why hasn’t anyone come back from the future and told us how to do it? There might be good reasons why it would be unwise to give us the secret of time travel at our present primitive state of development, but unless human nature changes radically, it is difficult to believe that some visitor from the future wouldn’t spill the beans. Of course, some people would claim that sightings of UFOs are evidence that we are being visited either by aliens or by people from the future. (If the aliens were to get here in reasonable time, they would need faster-than-light travel, so the two possibilities may be equivalent.) However, I think that any visit by aliens or people from the future would be much more obvious and, probably, much more unpleasant. If they are going to reveal themselves at all, why do so only to those who are not file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/i.html (3 of 5) [2/20/2001 3:15:45 AM]
A Brief History of Time - Stephen Hawking... Chapter 10 regarded as reliable witnesses? If they are trying to warn us of some great danger, they are not being very effective. A possible way to explain the absence of visitors from the future would be to say that the past is fixed because we have observed it and seen that it does not have the kind of warping needed to allow travel back from the future. On the other hand, the future is unknown and open, so it might well have the curvature required. This would mean that any time travel would be confined to the future. There would be no chance of Captain Kirk and the Starship Enterprise turning up at the present time. This might explain why we have not yet been overrun by tourists from the future, but it would not avoid the problems that would arise if one were able to go back and change history. Suppose, for example, you went back and killed your great-great-grandfather while he was still a child. There are many versions of this paradox but they are essentially equivalent: one would get contradictions if one were free to change the past. There seem to be two possible resolutions to the paradoxes posed by time travel. One I shall call the consistent histories approach. It says that even if space-time is warped so that it would be possible to travel into the past, what happens in space-time must be a consistent solution of the laws of physics. According to this viewpoint, you could not go back in time unless history showed that you had already arrived in the past and, while there, had not killed your great-great-grandfather or committed any other acts that would conflict with your current situation in the present. Moreover, when you did go back, you wouldn’t be able to change recorded history. That means you wouldn’t have free will to do what you wanted. Of course, one could say that free will is an illusion anyway. If there really is a complete unified theory that governs everything, it presumably also determines your actions. But it does so in a way that is impossible to calculate for an organism that is as complicated as a human being. The reason we say that humans have free will is because we can’t predict what they will do. However, if the human then goes off in a rocket ship and comes back before he or she set off, we will be able to predict what he or she will do because it will be part of recorded history. Thus, in that situation, the time traveler would have no free will. The other possible way to resolve the paradoxes of time travel might be called the alternative histories hypothesis. The idea here is that when time travelers go back to the past, they enter alternative histories which differ from recorded history. Thus they can act freely, without the constraint of consistency with their previous history. Steven Spiel-berg had fun with this notion in the Back to the Future films: Marty McFly was able to go back and change his parents’ courtship to a more satisfactory history. The alternative histories hypothesis sounds rather like Richard Feynman’s way of expressing quantum theory as a sum over histories, which was described in Chapters 4 and 8. This said that the universe didn’t just have a single history: rather it had every possible history, each with its own probability. However, there seems to be an important difference between Feynman’s proposal and alternative histories. In Feynman’s sum, each history comprises a complete space-time and everything in it. The space-time may be so warped that it is possible to travel in a rocket into the past. But the rocket would remain in the same space-time and therefore the same history, which would have to be consistent. Thus Feynman’s sum over histories proposal seems to support the consistent histories hypothesis rather than the alternative histories. The Feynman sum over histories does allow travel into the past on a microscopic scale. In Chapter 9 we saw that the laws of science are unchanged by combinations of the operations C, P, and T. This means that an antiparticle spinning in the anticlockwise direction and moving from A to B can also be viewed as an ordinary particle spinning clockwise and moving backward in time from B to A. Similarly, an ordinary particle moving forward in time is equivalent to an antiparticle moving backward in time. As has been discussed in this chapter and Chapter 7, “empty” space is filled with pairs of virtual particles and antiparticles that appear together, move apart, and then come back together and annihilate each other. So, one can regard the pair of particles as a single particle moving on a closed loop in space-time. When the pair is moving forward in time (from the event at which it appears to that at which it annihilates), it is called a particle. But when the particle is traveling back in time (from the event at which the pair annihilates to that at which it appears), it is said to be an antiparticle traveling forward in time. The explanation of how black holes can emit particles and radiation (given in Chapter 7) was that one member file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/i.html (4 of 5) [2/20/2001 3:15:45 AM]
A Brief History of Time - Stephen Hawking... Chapter 10 of a virtual particle/ antiparticle pair (say, the antiparticle) might fall into the black hole, leaving the other member without a partner with which to annihilate. The forsaken particle might fall into the hole as well, but it might also escape from the vicinity of the black hole. If so, to an observer at a distance it would appear to be a particle emitted by the black hole. One can, however, have a different but equivalent intuitive picture of the mechanism for emission from black holes. One can regard the member of the virtual pair that fell into the black hole (say, the antiparticle) as a particle traveling backward in time out of the hole. When it gets to the point at which the virtual particle/antiparticle pair appeared together, it is scattered by the gravitational field into a particle traveling forward in time and escaping from the black hole. If, instead, it were the particle member of the virtual pair that fell into the hole, one could regard it as an antiparticle traveling back in time and coming out of the black hole. Thus the radiation by black holes shows that quantum theory allows travel back in time on a microscopic scale and that such time travel can produce observable effects. One can therefore ask: does quantum theory allow time travel on a macroscopic scale, which people could use? At first sight, it seems it should. The Feynman sum over histories proposal is supposed to be over all histories. Thus it should include histories in which space-time is so warped that it is possible to travel into the past. Why then aren’t we in trouble with history? Suppose, for example, someone had gone back and given the Nazis the secret of the atom bomb? One would avoid these problems if what I call the chronology protection conjecture holds. This says that the laws of physics conspire to prevent macroscopic bodies from carrying information into the past. Like the cosmic censorship conjecture, it has not been proved but there are reasons to believe it is true. The reason to believe that chronology protection operates is that when space-time is warped enough to make travel into the past possible, virtual particles moving on closed loops in space-time can become real particles traveling forward in time at or below the speed of light. As these particles can go round the loop any number of times, they pass each point on their route many times. Thus their energy is counted over and over again and the energy density will become very large. This could give space-time a positive curvature that would not allow travel into the past. It is not yet clear whether these particles would cause positive or negative curvature or whether the curvature produced by some kinds of virtual particles might cancel that produced by other kinds. Thus the possibility of time travel remains open. But I’m not going to bet on it. My opponent might have the unfair advantage of knowing the future. PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/i.html (5 of 5) [2/20/2001 3:15:45 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 CHAPTER 11 THE UNIFICATION OF PHYSICS As was explained in the first chapter, it would be very difficult to construct a complete unified theory of everything in the universe all at one go. So instead we have made progress by finding partial theories that describe a limited range of happenings and by neglecting other effects or approximating them by certain numbers. (Chemistry, for example, allows us to calculate the interactions of atoms, without knowing the internal structure of an atom’s nucleus.) Ultimately, however, one would hope to find a complete, consistent, unified theory that would include all these partial theories as approximations, and that did not need to be adjusted to fit the facts by picking the values of certain arbitrary numbers in the theory. The quest for such a theory is known as “the unification of physics.” Einstein spent most of his later years unsuccessfully searching for a unified theory, but the time was not ripe: there were partial theories for gravity and the electromagnetic force, but very little was known about the nuclear forces. Moreover, Einstein refused to believe in the reality of quantum mechanics, despite the important role he had played in its development. Yet it seems that the uncertainty principle is a fundamental feature of the universe we live in. A successful unified theory must, therefore, necessarily incorporate this principle. As I shall describe, the prospects for finding such a theory seem to be much better now because we know so much more about the universe. But we must beware of overconfidence – we have had false dawns before! At the beginning of this century, for example, it was thought that everything could be explained in terms of the properties of continuous matter, such as elasticity and heat conduction. The discovery of atomic structure and the uncertainty principle put an emphatic end to that. Then again, in 1928, physicist and Nobel Prize winner Max Born told a group of visitors to Gottingen University, “Physics, as we know it, will be over in six months.” His confidence was based on the recent discovery by Dirac of the equation that governed the electron. It was thought that a similar equation would govern the proton, which was the only other particle known at the time, and that would be the end of theoretical physics. However, the discovery of the neutron and of nuclear forces knocked that one on the head too. Having said this, I still believe there are grounds for cautious optimism that we may now be near the end of the search for the ultimate laws of nature. In previous chapters I have described general relativity, the partial theory of gravity, and the partial theories that govern the weak, the strong, and the electromagnetic forces. The last three may be combined in so-called grand unified theories, or GUTs, which are not very satisfactory because they do not include gravity and because they contain a number of quantities, like the relative masses of different particles, that cannot be predicted from the theory but have to be chosen to fit observations. The main difficulty in finding a theory that unifies gravity with the other forces is that general relativity is a “classical” theory; that is, it does not incorporate the uncertainty principle of quantum mechanics. On the other hand, the other partial theories depend on quantum mechanics in an essential way. A necessary first step, therefore, is to combine general relativity with the uncertainty principle. As we have seen, this can produce some remarkable consequences, such as black holes not being black, and the universe not having any singularities but being completely self-contained and without a boundary. The trouble is, as explained in Chapter 7, that the uncertainty principle means that even “empty” space is filled with pairs of virtual particles and antiparticles. These pairs would have an infinite amount of energy and, therefore, by Einstein’s famous equation E = mc2, they would have an infinite amount of mass. Their gravitational attraction would thus curve up the universe to infinitely small size. Rather similar, seemingly absurd infinities occur in the other partial theories, but in all these cases the infinities can be canceled out by a process called renormalization. This involves canceling the infinities by introducing other infinities. Although this technique is rather dubious mathematically, it does seem to work in practice, and has been used with these theories to make predictions that agree with observations to an extraordinary degree of accuracy. Renormalization, however, does have a serious drawback from the point of view of trying to find a complete theory, because it means that the actual values of the masses and the strengths of the forces cannot be predicted from the theory, but have to be chosen to fit the observations. In attempting to incorporate the uncertainty principle into general relativity, one has only two quantities that can be adjusted: the strength of gravity and the value of the cosmological constant. But adjusting these is not file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (1 of 11) [2/20/2001 3:15:56 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 sufficient to remove all the infinities. One therefore has a theory that seems to predict that certain quantities, such as the curvature of space-time, are really infinite, yet these quantities can be observed and measured to be perfectly finite! This problem in combining general relativity and the uncertainty principle had been suspected for some time, but was finally confirmed by detailed calculations in 1972. Four years later, a possible solution, called “supergravity,” was suggested. The idea was to combine the spin-2 particle called the graviton, which carries the gravitational force, with certain other particles of spin 3/2, 1, ½, and 0. In a sense, all these particles could then be regarded as different aspects of the same “superparticle,” thus unifying the matter particles with spin ½ and 3/2 with the force-carrying particles of spin 0, 1, and 2. The virtual particle/antiparticle pairs of spin ½ and 3/2 would have negative energy, and so would tend to cancel out the positive energy of the spin 2, 1, and 0 virtual pairs. This would cause many of the possible infinities to cancel out, but it was suspected that some infinities might still remain. However, the calculations required to find out whether or not there were any infinities left uncancelled were so long and difficult that no one was prepared to undertake them. Even with a computer it was reckoned it would take at least four years, and the chances were very high that one would make at least one mistake, probably more. So one would know one had the right answer only if someone else repeated the calculation and got the same answer, and that did not seem very likely! Despite these problems, and the fact that the particles in the super-gravity theories did not seem to match the observed particles, most scientists believed that supergravity was probably the right answer to the problem of the unification of physics. It seemed the best way of unifying gravity with the other forces. However, in 1984 there was a remarkable change of opinion in favor of what are called string theories. In these theories the basic objects are not particles, which occupy a single point of space, but things that have a length but no other dimension, like an infinitely thin piece of string. These strings may have ends (the so-called open strings) or they may be joined up with themselves in closed loops (closed strings) Figure 11:1 and Figure 11:2. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (2 of 11) [2/20/2001 3:15:56 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 Figures 11:1 & 11:2 A particle occupies one point of space at each instant of time. Thus its history can be represented by a line in space-time (the “world-line”). A string, on the other hand, occupies a line in space at each moment of time. So its history in space-time is a two-dimensional surface called the world-sheet. (Any point on such a world-sheet can be described by two numbers, one specifying the time and the other the position of the point on the string.) The world-sheet of an open string is a strip: its edges represent the paths through space-time of the ends of the string Figure 11:1. The world-sheet of a closed string is a cylinder or tube Figure 11:2: a slice through the tube is a circle, which represents the position of the string at one particular time. Two pieces of string can join together to form a single string; in the case of open strings they simply join at the ends Figure 11:3, while in the case of closed strings it is like the two legs joining on a pair of trousers Figure 11:4. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (3 of 11) [2/20/2001 3:15:56 AM]
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A Brief History of Time - Stephen Hawking... Chapter 11 Figure 11:4 Similarly, a single piece of string can divide into two strings. In string theories, what were previously thought of as particles are now pictured as waves traveling down the string, like waves on a vibrating kite string. The emission or absorption of one particle by another corresponds to the dividing or joining together of strings. For example, the gravitational force of the sun on the earth was pictured in particle theories as being caused by the emission of a graviton by a particle in the sun and its absorption by a particle in the earth Figure 11:5. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (5 of 11) [2/20/2001 3:15:56 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 Figures 11:5 & 11:6 In string theory, this process corresponds to an H-shaped tube or pipe Figure 11:6 (string theory is rather like plumbing, in a way). The two vertical sides of the H correspond to the particles in the sun and the earth, and the horizontal crossbar corresponds to the graviton that travels between them. String theory has a curious history. It was originally invented in the late 1960s in an attempt to find a theory to describe the strong force. The idea was that particles like the proton and the neutron could be regarded as waves on a string. The strong forces between the particles would correspond to pieces of string that went between other bits of string, as in a spider’s web. For this theory to give the observed value of the strong force between particles, the strings had to be like rubber bands with a pull of about ten tons. In 1974 Joel Scherk from Paris and John Schwarz from the California Institute of Technology published a paper in which they showed that string theory could describe the gravitational force, but only if the tension in the string were very much higher, about a thousand million million million million million million tons (1 with thirty-nine zeros after it). The predictions of the string theory would be just the same as those of general relativity on normal length scales, but they would differ at very small distances, less than a thousand million million million million millionth of a centimeter (a centimeter divided by 1 with thirty-three zeros after it). Their work did not receive much attention, however, because at just about that time most people abandoned the original string theory of the strong force in favor of the theory based on quarks and gluons, which seemed to fit much better with observations. Scherk died in tragic circumstances (he suffered from diabetes and went into a coma when no one was around to give him an injection of insulin). So Schwarz was left alone as almost the only supporter of string theory, but now with the much higher proposed value of the string tension. In 1984 interest in strings suddenly revived, apparently for two reasons. One was that people were not really making much progress toward showing that supergravity was finite or that it could explain the kinds of particles that we observe. The other was the publication of a paper by John Schwarz and Mike Green of Queen Mary College, London, that showed that string theory might be able to explain the existence of particles that have a built-in left-handedness, like some of the particles that we observe. Whatever the reasons, a large number of people soon began to work on string theory and a new version was developed, the so-called heterotic string, which seemed as if it might be able to explain the types of particles that we observe. String theories also lead to infinities, but it is thought they will all cancel out in versions like the heterotic string file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (6 of 11) [2/20/2001 3:15:56 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 (though this is not yet known for certain). String theories, however, have a bigger problem: they seem to be consistent only if space-time has either ten or twenty-six dimensions, instead of the usual four! Of course, extra space-time dimensions are a commonplace of science fiction indeed, they provide an ideal way of overcoming the normal restriction of general relativity that one cannot travel faster than light or back in time (see Chapter 10). The idea is to take a shortcut through the extra dimensions. One can picture this in the following way. Imagine that the space we live in has only two dimensions and is curved like the surface of an anchor ring or torus Figure 11:7. Figure 11:7 If you were on one side of the inside edge of the ring and you wanted to get to a point on the other side, you would have to go round the inner edge of the ring. However, if you were able to travel in the third dimension, you could cut straight across. Why don’t we notice all these extra dimensions, if they are really there? Why do we see only three space dimensions and one time dimension? The suggestion is that the other dimensions are curved up into a space of very small size, something like a million million million million millionth of an inch. This is so small that we just don’t notice it: we see only one time dimension and three space dimensions, in which space-time is fairly flat. It is like the surface of a straw. If you look at it closely, you see it is two-dimensional (the position of a point on the straw is described by two numbers, the length along the straw and the distance round the circular direction). But if you look at it from a distance, you don’t see the thickness of the straw and it looks one-dimensional (the position of a point is specified only by the length along the straw). So it is with space-time: on a very small scale it is ten-dimensional and highly curved, but on bigger scales you don’t see the curvature or the extra dimensions. If this picture is correct, it spells bad news for would-be space travelers: the extra dimensions would be far too small to allow a spaceship through. However, it raises another major problem. Why should some, but not all, of the dimensions be curled up into a small ball? Presumably, in the very early universe all the dimensions would have been very curved. Why did one time dimension and three space dimensions flatten file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (7 of 11) [2/20/2001 3:15:56 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 out, while the other dimensions remain tightly curled up? One possible answer is the anthropic principle. Two space dimensions do not seem to be enough to allow for the development of complicated beings like us. For example, two-dimensional animals living on a one-dimensional earth would have to climb over each other in order to get past each other. If a two-dimensional creature ate something it could not digest completely, it would have to bring up the remains the same way it swallowed them, because if there were a passage right through its body, it would divide the creature into two separate halves: our two-dimensional being would fall apart Figure 11:8. Similarly, it is difficult to see how there could be any circulation of the blood in a two-dimensional creature. Figure 11:8 There would also be problems with more than three space dimensions. The gravitational force between two bodies would decrease more rapidly with distance than it does in three dimensions. (In three dimensions, the gravitational force drops to 1/4 if one doubles the distance. In four dimensions it would drop to 1/5, in five dimensions to 1/6, and so on.) The significance of this is that the orbits of planets, like the earth, around the sun would be unstable: the least disturbance from a circular orbit (such as would be caused by the gravitational attraction of other planets) would result in the earth spiraling away from or into the sun. We would either freeze or be burned up. In fact, the same behavior of gravity with distance in more than three space dimensions means that the sun would not be able to exist in a stable state with pressure balancing gravity. It would either fall apart or it would collapse to form a black hole. In either case, it would not be of much use as a source of heat and light for life on earth. On a smaller scale, the electrical forces that cause the electrons to orbit round the nucleus in an atom would behave in the same way as gravitational forces. Thus the electrons would either escape from the atom altogether or would spiral into the nucleus. In either case, one could not have atoms as we know them. It seems clear then that life, at least as we know it, can exist only in regions of space-time in which one time dimension and three space dimensions are not curled up small. This would mean that one could appeal to the weak anthropic principle, provided one could show that string theory does at least allow there to be such regions of the universe – and it seems that indeed string theory does. There may well be other regions of the universe, or other universes (whatever that may mean), in which all the dimensions are curled up small or in which more than four dimensions are nearly flat, but there would be no intelligent beings in such regions to file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (8 of 11) [2/20/2001 3:15:56 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 observe the different number of effective dimensions. Another problem is that there are at least four different string theories (open strings and three different closed string theories) and millions of ways in which the extra dimensions predicted by string theory could be curled up. Why should just one string theory and one kind of curling up be picked out? For a time there seemed no answer, and progress got bogged down. Then, from about 1994, people started discovering what are called dualities: different string theories and different ways of curling up the extra dimensions could lead to the same results in four dimensions. Moreover, as well as particles, which occupy a single point of space, and strings, which are lines, there were found to be other objects called p-branes, which occupied two-dimensional or higher-dimensional volumes in space. (A particle can be regarded as a 0-brane and a string as a 1-brane but there were also p-branes for p=2 to p=9.) What this seems to indicate is that there is a sort of democracy among supergravity, string, and p-brane theories: they seem to fit together but none can be said to be more fundamental than the others. They appear to be different approximations to some fundamental theory that are valid in different situations. People have searched for this underlying theory, but without any success so far. However, I believe there may not be any single formulation of the fundamental theory any more than, as Godel showed, one could formulate arithmetic in terms of a single set of axioms. Instead it may be like maps – you can’t use a single map to describe the surface of the earth or an anchor ring: you need at least two maps in the case of the earth and four for the anchor ring to cover every point. Each map is valid only in a limited region, but different maps will have a region of overlap. The collection of maps provides a complete description of the surface. Similarly, in physics it may be necessary to use different formulations in different situations, but two different formulations would agree in situations where they can both be applied. The whole collection of different formulations could be regarded as a complete unified theory, though one that could not be expressed in terms of a single set of postulates. But can there really be such a unified theory? Or are we perhaps just chasing a mirage? There seem to be three possibilities: 1. There really is a complete unified theory (or a collection of overlapping formulations), which we will someday discover if we are smart enough. 2. There is no ultimate theory of the universe, just an infinite sequence of theories that describe the universe more and more accurately. 3. There is no theory of the universe: events cannot be predicted beyond a certain extent but occur in a random and arbitrary manner. Some would argue for the third possibility on the grounds that if there were a complete set of laws, that would infringe God’s freedom to change his mind and intervene in the world. It’s a bit like the old paradox: can God make a stone so heavy that he can’t lift it? But the idea that God might want to change his mind is an example of the fallacy, pointed out by St. Augustine, of imagining God as a being existing in time: time is a property only of the universe that God created. Presumably, he knew what he intended when he set it up! With the advent of quantum mechanics, we have come to recognize that events cannot be predicted with complete accuracy but that there is always a degree of uncertainty. If one likes, one could ascribe this randomness to the intervention of God, but it would be a very strange kind of intervention: there is no evidence that it is directed toward any purpose. Indeed, if it were, it would by definition not be random. In modern times, we have effectively removed the third possibility above by redefining the goal of science: our aim is to formulate a set of laws that enables us to predict events only up to the limit set by the uncertainty principle. The second possibility, that there is an infinite sequence of more and more refined theories, is in agreement with all our experience so far. On many occasions we have increased the sensitivity of our measurements or made a new class of observations, only to discover new phenomena that were not predicted by the existing theory, and to account for these we have had to develop a more advanced theory. It would therefore not be very surprising if the present generation of grand unified theories was wrong in claiming that nothing essentially new will happen between the electroweak unification energy of about 100 GeV and the grand unification energy of about a thousand million million GeV. We might indeed expect to find several new layers of structure more file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (9 of 11) [2/20/2001 3:15:56 AM]
A Brief History of Time - Stephen Hawking... Chapter 11 basic than the quarks and electrons that we now regard as “elementary” particles. However, it seems that gravity may provide a limit to this sequence of “boxes within boxes.” If one had a particle with an energy above what is called the Planck energy, ten million million million GeV (1 followed by nineteen zeros), its mass would be so concentrated that it would cut itself off from the rest of the universe and form a little black hole. Thus it does seem that the sequence of more and more refined theories should have some limit as we go to higher and higher energies, so that there should be some ultimate theory of the universe. Of course, the Planck energy is a very long way from the energies of around a hundred GeV, which are the most that we can produce in the laboratory at the present time. We shall not bridge that gap with particle accelerators in the foreseeable future! The very early stages of the universe, however, are an arena where such energies must have occurred. I think that there is a good chance that the study of the early universe and the requirements of mathematical consistency will lead us to a complete unified theory within the lifetime of some of us who are around today, always presuming we don’t blow ourselves up first. What would it mean if we actually did discover the ultimate theory of the universe? As was explained in Chapter 1, we could never be quite sure that we had indeed found the correct theory, since theories can’t be proved. But if the theory was mathematically consistent and always gave predictions that agreed with observations, we could be reasonably confident that it was the right one. It would bring to an end a long and glorious chapter in the history of humanity’s intellectual struggle to understand the universe. But it would also revolutionize the ordinary person’s understanding of the laws that govern the universe. In Newton’s time it was possible for an educated person to have a grasp of the whole of human knowledge, at least in outline. But since then, the pace of the development of science has made this impossible. Because theories are always being changed to account for new observations, they are never properly digested or simplified so that ordinary people can understand them. You have to be a specialist, and even then you can only hope to have a proper grasp of a small proportion of the scientific theories. Further, the rate of progress is so rapid that what one learns at school or university is always a bit out of date. Only a few people can keep up with the rapidly advancing frontier of knowledge, and they have to devote their whole time to it and specialize in a small area. The rest of the population has little idea of the advances that are being made or the excitement they are generating. Seventy years ago, if Eddington is to be believed, only two people understood the general theory of relativity. Nowadays tens of thousands of university graduates do, and many millions of people are at least familiar with the idea. If a complete unified theory was discovered, it would only be a matter of time before it was digested and simplified in the same way and taught in schools, at least in outline. We would then all be able to have some understanding of the laws that govern the universe and are responsible for our existence. Even if we do discover a complete unified theory, it would not mean that we would be able to predict events in general, for two reasons. The first is the limitation that the uncertainty principle of quantum mechanics sets on our powers of prediction. There is nothing we can do to get around that. In practice, however, this first limitation is less restrictive than the second one. It arises from the fact that we could not solve the equations of the theory exactly, except in very simple situations. (We cannot even solve exactly for the motion of three bodies in Newton’s theory of gravity, and the difficulty increases with the number of bodies and the complexity of the theory.) We already know the laws that govern the behavior of matter under all but the most extreme conditions. In particular, we know the basic laws that underlie all of chemistry and biology. Yet we have certainly not reduced these subjects to the status of solved problems: we have, as yet, had little success in predicting human behavior from mathematical equations! So even if we do find a complete set of basic laws, there will still be in the years ahead the intellectually challenging task of developing better approximation methods, so that we can make useful predictions of the probable outcomes in complicated and realistic situations. A complete, consistent, unified theory is only the first step: our goal is a complete understanding of the events around us, and of our own existence. PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/j.html (10 of 11) [2/20/2001 3:15:56 AM]
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A Brief History of Time - Stephen Hawking... Chapter 12 CHAPTER 12 CONCLUSION We find ourselves in a bewildering world. We want to make sense of what we see around us and to ask: What is the nature of the universe? What is our place in it and where did it and we come from? Why is it the way it is? To try to answer these questions we adopt some “world picture.” Just as an infinite tower of tortoises supporting the fiat earth is such a picture, so is the theory of superstrings. Both are theories of the universe, though the latter is much more mathematical and precise than the former. Both theories lack observational evidence: no one has ever seen a giant tortoise with the earth on its back, but then, no one has seen a superstring either. However, the tortoise theory fails to be a good scientific theory because it predicts that people should be able to fall off the edge of the world. This has not been found to agree with experience, unless that turns out to be the explanation for the people who are supposed to have disappeared in the Bermuda Triangle! The earliest theoretical attempts to describe and explain the universe involved the idea that events and natural phenomena were controlled by spirits with human emotions who acted in a very humanlike and unpredictable manner. These spirits inhabited natural objects, like rivers and mountains, including celestial bodies, like the sun and moon. They had to be placated and their favor sought in order to ensure the fertility of the soil and the rotation of the seasons. Gradually, however, it must have been noticed that there were certain regularities: the sun always rose in the east and set in the west, whether or not a sacrifice had been made to the sun god. Further, the sun, the moon, and the planets followed precise paths across the sky that could be predicted in advance with considerable accuracy. The sun and the moon might still be gods, but they were gods who obeyed strict laws, apparently without any exceptions, if one discounts stories like that of the sun stopping for Joshua. At first, these regularities and laws were obvious only in astronomy and a few other situations. However, as civilization developed, and particularly in the last 300 years, more and more regularities and laws were discovered. The success of these laws led Laplace at the beginning of the nineteenth century to postulate scientific determinism; that is, he suggested that there would be a set of laws that would determine the evolution of the universe precisely, given its configuration at one time. Laplace’s determinism was incomplete in two ways. It did not say how the laws should be chosen and it did not specify the initial configuration of the universe. These were left to God. God would choose how the universe began and what laws it obeyed, but he would not intervene in the universe once it had started. In effect, God was confined to the areas that nineteenth-century science did not understand. We now know that Laplace’s hopes of determinism cannot be realized, at least in the terms he had in mind. The uncertainty principle of quantum mechanics implies that certain pairs of quantities, such as the position and velocity of a particle, cannot both be predicted with complete accuracy. Quantum mechanics deals with this situation via a class of quantum theories in which particles don’t have well-defined positions and velocities but are represented by a wave. These quantum theories are deterministic in the sense that they give laws for the evolution of the wave with time. Thus if one knows the wave at one time, one can calculate it at any other time. The unpredictable, random element comes in only when we try to interpret the wave in terms of the positions and velocities of particles. But maybe that is our mistake: maybe there are no particle positions and velocities, but only waves. It is just that we try to fit the waves to our preconceived ideas of positions and velocities. The resulting mismatch is the cause of the apparent unpredictability. In effect, we have redefined the task of science to be the discovery of laws that will enable us to predict events up to the limits set by the uncertainty principle. The question remains, however: how or why were the laws and the initial state of the universe chosen? In this book I have given special prominence to the laws that govern gravity, because it is gravity that shapes the large-scale structure of the universe, even though it is the weakest of the four categories of forces. The laws of gravity were incompatible with the view held until quite recently that the universe is unchanging in time: file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/k.html (1 of 4) [2/20/2001 3:16:08 AM]
A Brief History of Time - Stephen Hawking... Chapter 12 the fact that gravity is always attractive implies that the universe must be either expanding or contracting. According to the general theory of relativity, there must have been a state of infinite density in the past, the big bang, which would have been an effective beginning of time. Similarly, if the whole universe recollapsed, there must be another state of infinite density in the future, the big crunch, which would be an end of time. Even if the whole universe did not recollapse, there would be singularities in any localized regions that collapsed to form black holes. These singularities would be an end of time for anyone who fell into the black hole. At the big bang and other singularities, all the laws would have broken down, so God would still have had complete freedom to choose what happened and how the universe began. When we combine quantum mechanics with general relativity, there seems to be a new possibility that did not arise before: that space and time together might form a finite, four-dimensional space without singularities or boundaries, like the surface of the earth but with more dimensions. It seems that this idea could explain many of the observed features of the universe, such as its large-scale uniformity and also the smaller-scale departures from homogeneity, like galaxies, stars, and even human beings. It could even account for the arrow of time that we observe. But if the universe is completely self-contained, with no singularities or boundaries, and completely described by a unified theory, that has profound implications for the role of God as Creator. Einstein once asked the question: “How much choice did God have in constructing the universe?” If the no boundary proposal is correct, he had no freedom at all to choose initial conditions. He would, of course, still have had the freedom to choose the laws that the universe obeyed. This, however, may not really have been all that much of a choice; there may well be only one, or a small number, of complete unified theories, such as the heterotic string theory, that are self-consistent and allow the existence of structures as complicated as human beings who can investigate the laws of the universe and ask about the nature of God. Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, if so, does he have any other effect on the universe? And who created him? Up to now, most scientists have been too occupied with the development of new theories that describe what the universe is to ask the question why. On the other hand, the people whose business it is to ask why, the philosophers, have not been able to keep up with the advance of scientific theories. In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, “The sole remaining task for philosophy is the analysis of language.” What a comedown from the great tradition of philosophy from Aristotle to Kant! However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason – for then we would know the mind of God. ALBERT EINSTEIN Einstein’s connection with the politics of the nuclear bomb is well known: he signed the famous letter to President Franklin Roosevelt that persuaded the United States to take the idea seriously, and he engaged in postwar efforts to prevent nuclear war. But these were not just the isolated actions of a scientist dragged into the world of politics. Einstein’s life was, in fact, to use his own words, “divided between politics and equations.” Einstein’s earliest political activity came during the First World War, when he was a professor in Berlin. Sickened by what he saw as the waste of human lives, he became involved in antiwar demonstrations. His file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/k.html (2 of 4) [2/20/2001 3:16:08 AM]
A Brief History of Time - Stephen Hawking... Chapter 12 advocacy of civil disobedience and public encouragement of people to refuse conscription did little to endear him to his colleagues. Then, following the war, he directed his efforts toward reconciliation and improving international relations. This too did not make him popular, and soon his politics were making it difficult for him to visit the United States, even to give lectures. Einstein’s second great cause was Zionism. Although he was Jewish by descent, Einstein rejected the biblical idea of God. However, a growing awareness of anti-Semitism, both before and during the First World War, led him gradually to identify with the Jewish community, and later to become an outspoken supporter of Zionism. Once more unpopularity did not stop him from speaking his mind. His theories came under attack; an anti-Einstein organization was even set up. One man was convicted of inciting others to murder Einstein (and fined a mere six dollars). But Einstein was phlegmatic. When a book was published entitled 100 Authors Against Einstein, he retorted, “If I were wrong, then one would have been enough!” In 1933, Hitler came to power. Einstein was in America, and declared he would not return to Germany. Then, while Nazi militia raided his house and confiscated his bank account, a Berlin newspaper displayed the headline “Good News from Einstein – He’s Not Coming Back.” In the face of the Nazi threat, Einstein renounced pacifism, and eventually, fearing that German scientists would build a nuclear bomb, proposed that the United States should develop its own. But even before the first atomic bomb had been detonated, he was publicly warning of the dangers of nuclear war and proposing international control of nuclear weaponry. Throughout his life, Einstein’s efforts toward peace probably achieved little that would last – and certainly won him few friends. His vocal support of the Zionist cause, however, was duly recognized in 1952, when he was offered the presidency of Israel. He declined, saying he thought he was too naive in politics. But perhaps his real reason was different: to quote him again, “Equations are more important to me, because politics is for the present, but an equation is something for eternity.” GALILEO GALILEI Galileo, perhaps more than any other single person, was responsible for the birth of modern science. His renowned conflict with the Catholic Church was central to his philosophy, for Galileo was one of the first to argue that man could hope to understand how the world works, and, moreover, that we could do this by observing the real world. Galileo had believed Copernican theory (that the planets orbited the sun) since early on, but it was only when he found the evidence needed to support the idea that he started to publicly support it. He wrote about Copernicus’s theory in Italian (not the usual academic Latin), and soon his views became widely supported outside the universities. This annoyed the Aristotelian professors, who united against him seeking to persuade the Catholic Church to ban Copernicanism. Galileo, worried by this, traveled to Rome to speak to the ecclesiastical authorities. He argued that the Bible was not intended to tell us anything about scientific theories, and that it was usual to assume that, where the Bible conflicted with common sense, it was being allegorical. But the Church was afraid of a scandal that might undermine its fight against Protestantism, and so took repressive measures. It declared Copernicanism “false and erroneous” in 1616, and commanded Galileo never again to “defend or hold” the doctrine. Galileo acquiesced. In 1623, a longtime friend of Galileo’s became the Pope. Immediately Galileo tried to get the 1616 decree revoked. He failed, but he did manage to get permission to write a book discussing both Aristotelian and Copernican theories, on two conditions: he would not take sides and would come to the conclusion that man could in any case not determine how the world worked because God could bring about the same effects in ways unimagined by man, who could not place restrictions on God’s omnipotence. The book, Dialogue Concerning the Two Chief World Systems, was completed and published in 1632, with the full backing of the censors – and was immediately greeted throughout Europe as a literary and philosophical masterpiece. Soon the Pope, realizing that people were seeing the book as a convincing argument in favor of file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/k.html (3 of 4) [2/20/2001 3:16:08 AM]
A Brief History of Time - Stephen Hawking... Chapter 12 Copernicanism, regretted having allowed its publication. The Pope argued that although the book had the official blessing of the censors, Galileo had nevertheless contravened the 1616 decree. He brought Galileo before the Inquisition, who sentenced him to house arrest for life and commanded him to publicly renounce Copernicanism. For a second time, Galileo acquiesced. Galileo remained a faithful Catholic, but his belief in the independence of science had not been crushed. Four years before his death in 1642, while he was still under house arrest, the manuscript of his second major book was smuggled to a publisher in Holland. It was this work, referred to as Two New Sciences, even more than his support for Copernicus, that was to be the genesis of modern physics. ISAAC NEWTON Isaac Newton was not a pleasant man. His relations with other academics were notorious, with most of his later life spent embroiled in heated disputes. Following publication of Principia Mathematica – surely the most influential book ever written in physics – Newton had risen rapidly into public prominence. He was appointed president of the Royal Society and became the first scientist ever to be knighted. Newton soon clashed with the Astronomer Royal, John Flamsteed, who had earlier provided Newton with much-needed data for Principia, but was now withholding information that Newton wanted. Newton would not take no for an answer: he had himself appointed to the governing body of the Royal Observatory and then tried to force immediate publication of the data. Eventually he arranged for Flamsteed’s work to be seized and prepared for publication by Flamsteed’s mortal enemy, Edmond Halley. But Flamsteed took the case to court and, in the nick of time, won a court order preventing distribution of the stolen work. Newton was incensed and sought his revenge by systematically deleting all references to Flamsteed in later editions of Principia. A more serious dispute arose with the German philosopher Gottfried Leibniz. Both Leibniz and Newton had independently developed a branch of mathematics called calculus, which underlies most of modern physics. Although we now know that Newton discovered calculus years before Leibniz, he published his work much later. A major row ensued over who had been first, with scientists vigorously defending both contenders. It is remarkable, however, that most of the articles appearing in defense of Newton were originally written by his own hand – and only published in the name of friends! As the row grew, Leibniz made the mistake of appealing to the Royal Society to resolve the dispute. Newton, as president, appointed an “impartial” committee to investigate, coincidentally consisting entirely of Newton’s friends! But that was not all: Newton then wrote the committee’s report himself and had the Royal Society publish it, officially accusing Leibniz of plagiarism. Still unsatisfied, he then wrote an anonymous review of the report in the Royal Society’s own periodical. Following the death of Leibniz, Newton is reported to have declared that he had taken great satisfaction in “breaking Leibniz’s heart.” During the period of these two disputes, Newton had already left Cambridge and academe. He had been active in anti-Catholic politics at Cambridge, and later in Parliament, and was rewarded eventually with the lucrative post of Warden of the Royal Mint. Here he used his talents for deviousness and vitriol in a more socially acceptable way, successfully conducting a major campaign against counterfeiting, even sending several men to their death on the gallows. PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/k.html (4 of 4) [2/20/2001 3:16:08 AM]
A Brief History of Time - Stephen Hawking... Glossary GLOSSARY Absolute zero: The lowest possible temperature, at which substances contain no heat energy. Acceleration: The rate at which the speed of an object is changing. Anthropic principle: We see the universe the way it is because if it were different we would not be here to observe it. Antiparticle: Each type of matter particle has a corresponding antiparticle. When a particle collides with its antiparticle, they annihilate, leaving only energy. Atom: The basic unit of ordinary matter, made up of a tiny nucleus (consisting of protons and neutrons) surrounded by orbiting electrons. Big bang: The singularity at the beginning of the universe. Big crunch: The singularity at the end of the universe. Black hole: A region of space-time from which nothing, not even light, can escape, because gravity is so strong. Casimir effect: The attractive pressure between two flat, parallel metal plates placed very near to each other in a vacuum. The pressure is due to a reduction in the usual number of virtual particles in the space between the plates. Chandrasekhar limit: The maximum possible mass of a stable cold star, above which it must collapse into a black hole. Conservation of energy: The law of science that states that energy (or its equivalent in mass) can neither be created nor destroyed. Coordinates: Numbers that specify the position of a point in space and time. Cosmological constant: A mathematical device used by Einstein to give space-time an inbuilt tendency to expand. Cosmology: The study of the universe as a whole. Dark matter: Matter in galaxies, clusters, and possibly between clusters, that can not be observed directly but can be detected by its gravitational effect. As much as 90 percent of the mass of the universe may be in the form of dark matter. Duality: A correspondence between apparently different theories that lead to the same physical results. Einstein-Rosen bridge: A thin tube of space-time linking two black holes. Also see Wormhole. Electric charge: A property of a particle by which it may repel (or attract) other particles that have a charge of similar (or opposite) sign. Electromagnetic force: The force that arises between particles with electric charge; the second strongest of the four fundamental forces. Electron: A particle with negative electric charge that orbits the nucleus of an atom. Electroweak unification energy: The energy (around 100 GeV) above which the distinction between the electromagnetic force and the weak force disappears. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/l.html (1 of 4) [2/20/2001 3:16:19 AM]
A Brief History of Time - Stephen Hawking... Glossary Elementary particle: A particle that, it is believed, cannot be subdivided. Event: A point in space-time, specified by its time and place. Event horizon: The boundary of a black hole. Exclusion principle: The idea that two identical spin-1/2 particles cannot have (within the limits set by the uncertainty principle) both the same position and the same velocity. Field: Something that exists throughout space and time, as opposed to a particle that exists at only one point at a time. Frequency: For a wave, the number of complete cycles per second. Gamma rays: Electromagnetic rays of very short wavelength, produced in radio-active decay or by collisions of elementary particles. General relativity: Einstein’s theory based on the idea that the laws of science should be the same for all observers, no matter how they are moving. It explains the force of gravity in terms of the curvature of a four-dimensional space-time. Geodesic: The shortest (or longest) path between two points. Grand unification energy: The energy above which, it is believed, the electro-magnetic force, weak force, and strong force become indistinguishable from each other. Grand unified theory (GUT): A theory which unifies the electromagnetic, strong, and weak forces. Imaginary time: Time measured using imaginary numbers. Light cone: A surface in space-time that marks out the possible directions for light rays passing through a given event. Light-second (light-year): The distance traveled by light in one second (year). Magnetic field: The field responsible for magnetic forces, now incorporated along with the electric field, into the electromagnetic field. Mass: The quantity of matter in a body; its inertia, or resistance to acceleration. Microwave background radiation: The radiation from the glowing of the hot early universe, now so greatly red-shifted that it appears not as light but as microwaves (radio waves with a wavelength of a few centimeters). Also see COBE, on page 145. Naked singularity: A space-time singularity not surrounded by a black hole. Neutrino: An extremely light (possibly massless) particle that is affected only by the weak force and gravity. Neutron: An uncharged particle, very similar to the proton, which accounts for roughly half the particles in an atomic nucleus. Neutron star: A cold star, supported by the exclusion principle repulsion between neutrons. No boundary condition: The idea that the universe is finite but has no boundary (in imaginary time). Nuclear fusion: The process by which two nuclei collide and coalesce to form a single, heavier nucleus. Nucleus: The central part of an atom, consisting only of protons and neutrons, held together by the strong file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/l.html (2 of 4) [2/20/2001 3:16:19 AM]
A Brief History of Time - Stephen Hawking... Glossary force. Particle accelerator: A machine that, using electromagnets, can accelerate moving charged particles, giving them more energy. Phase: For a wave, the position in its cycle at a specified time: a measure of whether it is at a crest, a trough, or somewhere in between. Photon: A quantum of light. Planck’s quantum principle: The idea that light (or any other classical waves) can be emitted or absorbed only in discrete quanta, whose energy is proportional to their wavelength. Positron: The (positively charged) antiparticle of the electron. Primordial black hole: A black hole created in the very early universe. Proportional: ‘X is proportional to Y’ means that when Y is multiplied by any number, so is X. ‘X is inversely proportional to Y’ means that when Y is multiplied by any number, X is divided by that number. Proton: A positively charged particle, very similar to the neutron, that accounts for roughly half the particles in the nucleus of most atoms. Pulsar: A rotating neutron star that emits regular pulses of radio waves. Quantum: The indivisible unit in which waves may be emitted or absorbed. Quantum chromodynamics (QCD): The theory that describes the interactions of quarks and gluons. Quantum mechanics: The theory developed from Planck’s quantum principle and Heisenberg’s uncertainty principle. Quark: A (charged) elementary particle that feels the strong force. Protons and neutrons are each composed of three quarks. Radar: A system using pulsed radio waves to detect the position of objects by measuring the time it takes a single pulse to reach the object and be reflected back. Radioactivity: The spontaneous breakdown of one type of atomic nucleus into another. Red shift: The reddening of light from a star that is moving away from us, due to the Doppler effect. Singularity: A point in space-time at which the space-time curvature becomes infinite. Singularity theorem: A theorem that shows that a singularity must exist under certain circumstances – in particular, that the universe must have started with a singularity. Space-time: The four-dimensional space whose points are events. Spatial dimension: Any of the three dimensions that are spacelike – that is, any except the time dimension. Special relativity: Einstein’s theory based on the idea that the laws of science should be the same for all observers, no matter how they are moving, in the absence of gravitational phenomena. Spectrum: The component frequencies that make up a wave. The visible part of the sun’s spectrum can be seen in a rainbow. Spin: An internal property of elementary particles, related to, but not identical to, the everyday concept of spin. file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/l.html (3 of 4) [2/20/2001 3:16:19 AM]
A Brief History of Time - Stephen Hawking... Glossary Stationary state: One that is not changing with time: a sphere spinning at a constant rate is stationary because it looks identical at any given instant. String theory: A theory of physics in which particles are described as waves on strings. Strings have length but no other dimension. Strong force: The strongest of the four fundamental forces, with the shortest range of all. It holds the quarks together within protons and neutrons, and holds the protons and neutrons together to form atoms. Uncertainty principle: The principle, formulated by Heisenberg, that one can never be exactly sure of both the position and the velocity of a particle; the more accurately one knows the one, the less accurately one can know the other. Virtual particle: In quantum mechanics, a particle that can never be directly detected, but whose existence does have measurable effects. Wave/particle duality: The concept in quantum mechanics that there is no distinction between waves and particles; particles may sometimes behave like waves, and waves like particles. Wavelength: For a wave, the distance between two adjacent troughs or two adjacent crests. Weak force: The second weakest of the four fundamental forces, with a very short range. It affects all matter particles, but not force-carrying particles. Weight: The force exerted on a body by a gravitational field. It is proportional to, but not the same as, its mass. White dwarf: A stable cold star, supported by the exclusion principle repulsion between electrons. Wormhole: A thin tube of space-time connecting distant regions of the universe. Wormholes might also link to parallel or baby universes and could provide the possibility of time travel. PREVIOUS NEXT file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of time/l.html (4 of 4) [2/20/2001 3:16:19 AM]
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